On 2025-11-01 13:50:53 +0000, olcott said:
On 11/1/2025 4:10 AM, Mikko wrote:
On 2025-10-31 12:09:48 +0000, olcott said:
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote:Your words have much less material than all human >>>>>>>>>>>>>>>>>>> knowledge without
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that nobody >>>>>>>>>>>>>>>>>>>>>>>>>>> has verified all of
The conventional way to do proofs concludes >>>>>>>>>>>>>>>>>>>>>>>>>>>> thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese
this proves that Donald Trump is the Lord >>>>>>>>>>>>>>>>>>>>>>>>>>>> and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist.
Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>>>> some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of >>>>>>>>>>>>>>>>>>>>>>>> human
knowledge that can be expressed in language that >>>>>>>>>>>>>>>>>>>>>>>> begins
with a consistent set of basic facts and only >>>>>>>>>>>>>>>>>>>>>>>> applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that >>>>>>>>>>>>>>>>>>> you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess >>>>>>>>>>>>>>>>>> of Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of >>>>>>>>>>>>> the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted >>>>>>>>>>>> mess:
G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was >>>>>>>>>>> important to
cover languages that have no symbol for ':=' or '⊬'.
But that is irrelevant to my preference for the numbering >>>>>>>>>>> method.
His result is supposed to apply to any formal language
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any >>>>>>>>> symbols other than those of PA. A proof that requires other >>>>>>>>> symbols
proves nothing about PA.
If he started with Minimal Type Theory and the understanding
that cycles in the evaluation sequence of expressions indicate >>>>>>>> that they are semantically unsound we would have never heard
of Incompleteness.
Sooner or later someone would have found incompleteness anyway.
Incompleteness is a misconception nothing more.
No, it is a part of analytic knowledge. You have not shown any error >>>>> in Gödel's and other proofs that every theory of natural numbers is >>>>> incomplete. All you have shown is that you don't like the fact and
some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if
MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
Irrelevant to my comment that some parts of the theory are not yet
shown.
Its exactly the same thing a FOL with additional operators
to make it much more expressive. For example it can directly
expression any HOL logic expression.
You mean it includes "tertium non datur" and "the principle of explosion"?
I give an example.
Thank's but the definition of the theory is still missing.
On 2025-11-01 13:52:41 +0000, olcott said:
On 11/1/2025 4:16 AM, Mikko wrote:
On 2025-10-31 12:18:16 +0000, olcott said:
On 10/31/2025 6:10 AM, Mikko wrote:
On 2025-10-30 13:01:34 +0000, olcott said:
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human >>>>>>>>>>>>>>>>>>>> knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>>>>> some reasning that
When we require that correct reasoning is >>>>>>>>>>>>>>>>>>>>>>>>>>> only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that nobody >>>>>>>>>>>>>>>>>>>>>>>>>>>> has verified all of
The conventional way to do proofs concludes >>>>>>>>>>>>>>>>>>>>>>>>>>>>> thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from >>>>>>>>>>>>>>>>>>>>>>>>>>>>> green
cheese and the Moon is not made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese
this proves that Donald Trump is the Lord >>>>>>>>>>>>>>>>>>>>>>>>>>>>> and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly >>>>>>>>>>>>>>>>>>>>>>>>> exist
that can show there are any gaps in the body of >>>>>>>>>>>>>>>>>>>>>>>>> human
knowledge that can be expressed in language >>>>>>>>>>>>>>>>>>>>>>>>> that begins
with a consistent set of basic facts and only >>>>>>>>>>>>>>>>>>>>>>>>> applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that >>>>>>>>>>>>>>>>>>>> you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess >>>>>>>>>>>>>>>>>>> of Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too >>>>>>>>>>>>>>>>> inexpressive.
Natural numbers are one of the best understood topics of >>>>>>>>>>>>>>>> mathematics.
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of >>>>>>>>>>>>>> the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be >>>>>>>>>>>>> prime, it uses prime numbers to number the symbols, and >>>>>>>>>>>>> prime powers for the position.
And consequently Gödel's example of an undecidable formula >>>>>>>>>>>> is more complex
than needed. Which doesn't really matter as the complex >>>>>>>>>>>> details can be
hidden behind definitions.
...We are therefore confronted with a proposition which >>>>>>>>>>> asserts its own unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic.
G asserts it isn't provable, but asserting that there can be >>>>>>>>>> no number which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time >>>>>>>> is it?"
You cant say that in the language of Peano arithmetic.
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
PA can express everything we need to know about natural numbers.
Complexities can be hidden behind definitions.
MTT can express anything that we way to know about
formal expressions of language.
Since Incompleteness applies to PA OR ABOVE.
I go with above.
You may need to rewrite the proof for the language of the theory you >>>>> want to use, especially if you don't want to use the ordinary logic.
All that I did was get rid of the purely extraneous
complexity of Gödel's G
All that complexity is in Gödel's proof. You need to rewrite the proof
and verify its equivalence to the original if you want to ger rid of
the complexity.
But the complexity affects only the verification of the correctness of
the proof. The meaning of the conclusion is simple.
*Gödel already said that it is equivalent*
Not about my comment.
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote:Your words have much less material than all human >>>>>>>>>>>>>>>>>>> knowledge without
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that nobody >>>>>>>>>>>>>>>>>>>>>>>>>>> has verified all of
The conventional way to do proofs concludes >>>>>>>>>>>>>>>>>>>>>>>>>>>> thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese
this proves that Donald Trump is the Lord >>>>>>>>>>>>>>>>>>>>>>>>>>>> and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist.
Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>>>> some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of >>>>>>>>>>>>>>>>>>>>>>>> human
knowledge that can be expressed in language that >>>>>>>>>>>>>>>>>>>>>>>> begins
with a consistent set of basic facts and only >>>>>>>>>>>>>>>>>>>>>>>> applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that >>>>>>>>>>>>>>>>>>> you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess >>>>>>>>>>>>>>>>>> of Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of >>>>>>>>>>>>> the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be >>>>>>>>>>>> prime, it uses prime numbers to number the symbols, and >>>>>>>>>>>> prime powers for the position.
And consequently Gödel's example of an undecidable formula is >>>>>>>>>>> more complex
than needed. Which doesn't really matter as the complex >>>>>>>>>>> details can be
hidden behind definitions.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica >>>>>>>>>> And Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is >>>>>>>> incorrect.
Gödel's G when boiled down to its barest essence is simply
semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a
simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual >>>>> meaning of what I said is that you don't disagree with what I said in >>>>> my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
*Gödel already said that it is equivalent*
That doesn't mean 'equally complex'.
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not
ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in different ways.
--
Tristan Wibberley
The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not
ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
On 11/2/2025 6:41 AM, Mikko wrote:
On 2025-11-01 13:50:53 +0000, olcott said:
On 11/1/2025 4:10 AM, Mikko wrote:
On 2025-10-31 12:09:48 +0000, olcott said:Its exactly the same thing a FOL with additional operators
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:Incompleteness is a misconception nothing more.
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:His result is supposed to apply to any formal language
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted mess: >>>>>>>>>>>>> G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was important to
cover languages that have no symbol for ':=' or '⊬'. >>>>>>>>>>>>
But that is irrelevant to my preference for the numbering method. >>>>>>>>>>>
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any >>>>>>>>>> symbols other than those of PA. A proof that requires other symbols >>>>>>>>>> proves nothing about PA.
If he started with Minimal Type Theory and the understanding >>>>>>>>> that cycles in the evaluation sequence of expressions indicate >>>>>>>>> that they are semantically unsound we would have never heard >>>>>>>>> of Incompleteness.
Sooner or later someone would have found incompleteness anyway. >>>>>>>
No, it is a part of analytic knowledge. You have not shown any error >>>>>> in Gödel's and other proofs that every theory of natural numbers is >>>>>> incomplete. All you have shown is that you don't like the fact and >>>>>> some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if >>>>>> MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
Irrelevant to my comment that some parts of the theory are not yet shown. >>>
to make it much more expressive. For example it can directly
expression any HOL logic expression.
You mean it includes "tertium non datur" and "the principle of explosion"? >>
I give an example.
Thank's but the definition of the theory is still missing.
The theory is entirely expressed by the conventional
meaning of it operators and the syntax that I provided.
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not
ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
On 11/2/2025 6:43 AM, Mikko wrote:
On 2025-11-01 13:52:41 +0000, olcott said:
On 11/1/2025 4:16 AM, Mikko wrote:
On 2025-10-31 12:18:16 +0000, olcott said:
On 10/31/2025 6:10 AM, Mikko wrote:
On 2025-10-30 13:01:34 +0000, olcott said:
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheeseOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human
knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive.
Natural numbers are one of the best understood topics of mathematics.
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic. >>>>>>>>>>>
G asserts it isn't provable, but asserting that there can be no number
which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time is it?"
You cant say that in the language of Peano arithmetic.
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
PA can express everything we need to know about natural numbers.
Complexities can be hidden behind definitions.
MTT can express anything that we way to know about
formal expressions of language.
All that I did was get rid of the purely extraneousSince Incompleteness applies to PA OR ABOVE.
I go with above.
You may need to rewrite the proof for the language of the theory you >>>>>> want to use, especially if you don't want to use the ordinary logic. >>>>>
complexity of Gödel's G
All that complexity is in Gödel's proof. You need to rewrite the proof >>>> and verify its equivalence to the original if you want to ger rid of
the complexity.
But the complexity affects only the verification of the correctness of >>>> the proof. The meaning of the conclusion is simple.
*Gödel already said that it is equivalent*
Not about my comment.
I am not going to do any useless busy work.
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And >>>>>>>>>>> Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is simply semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual >>>>>> meaning of what I said is that you don't disagree with what I said in >>>>>> my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its own unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
On 2025-11-03 23:40:06 +0000, olcott said:
On 11/2/2025 6:41 AM, Mikko wrote:
On 2025-11-01 13:50:53 +0000, olcott said:
On 11/1/2025 4:10 AM, Mikko wrote:
On 2025-10-31 12:09:48 +0000, olcott said:
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:Incompleteness is a misconception nothing more.
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human >>>>>>>>>>>>>>>>>>>>> knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts >>>>>>>>>>>>>>>>>>>>>>>>> and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>>>>>> those facts and can
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>>>>>> some reasning that
When we require that correct reasoning is >>>>>>>>>>>>>>>>>>>>>>>>>>>> only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that nobody >>>>>>>>>>>>>>>>>>>>>>>>>>>>> has verified all of
The conventional way to do proofs >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concludes thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> green
cheese and the Moon is not made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese
this proves that Donald Trump is the Lord >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly >>>>>>>>>>>>>>>>>>>>>>>>>> exist
that can show there are any gaps in the body >>>>>>>>>>>>>>>>>>>>>>>>>> of human
knowledge that can be expressed in language >>>>>>>>>>>>>>>>>>>>>>>>>> that begins
with a consistent set of basic facts and only >>>>>>>>>>>>>>>>>>>>>>>>>> applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that >>>>>>>>>>>>>>>>>>>>> you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess >>>>>>>>>>>>>>>>>>>> of Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too >>>>>>>>>>>>>>>>>> inexpressive.
Natural numbers are one of the best understood topics >>>>>>>>>>>>>>>>> of mathematics.
Therefore restricting the assumptions to natural >>>>>>>>>>>>>>>>> numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims >>>>>>>>>>>>>>>>> and proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some >>>>>>>>>>>>>>> of the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted >>>>>>>>>>>>>> mess:
G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was >>>>>>>>>>>>> important to
cover languages that have no symbol for ':=' or '⊬'. >>>>>>>>>>>>>
But that is irrelevant to my preference for the numbering >>>>>>>>>>>>> method.
His result is supposed to apply to any formal language >>>>>>>>>>>> as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not >>>>>>>>>>> assume any
symbols other than those of PA. A proof that requires other >>>>>>>>>>> symbols
proves nothing about PA.
If he started with Minimal Type Theory and the understanding >>>>>>>>>> that cycles in the evaluation sequence of expressions indicate >>>>>>>>>> that they are semantically unsound we would have never heard >>>>>>>>>> of Incompleteness.
Sooner or later someone would have found incompleteness anyway. >>>>>>>>
No, it is a part of analytic knowledge. You have not shown any error >>>>>>> in Gödel's and other proofs that every theory of natural numbers is >>>>>>> incomplete. All you have shown is that you don't like the fact and >>>>>>> some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if >>>>>>> MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
Irrelevant to my comment that some parts of the theory are not yet
shown.
Its exactly the same thing a FOL with additional operators
to make it much more expressive. For example it can directly
expression any HOL logic expression.
You mean it includes "tertium non datur" and "the principle of
explosion"?
I give an example.
Thank's but the definition of the theory is still missing.
The theory is entirely expressed by the conventional
meaning of it operators and the syntax that I provided.
They don't express any theory. Shall we conclude that there is no theory?
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not
ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
The library predicate
unify_with_occurs_check is required fail if the unification would produce
a loop as a special feature of that feature. It is there for programmers
that want to ensure that they don't create loops were they don't want
loops. A loop in a data structure is a syntactic feature. There is no semantic considerations in unify_with_occurs_check.
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human >>>>>>>>>>>>>>>>>>>>> knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts >>>>>>>>>>>>>>>>>>>>>>>>> and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>>>>>> those facts and can
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>>>>>> some reasning that
When we require that correct reasoning is >>>>>>>>>>>>>>>>>>>>>>>>>>>> only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that nobody >>>>>>>>>>>>>>>>>>>>>>>>>>>>> has verified all of
The conventional way to do proofs >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concludes thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> green
cheese and the Moon is not made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese
this proves that Donald Trump is the Lord >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly >>>>>>>>>>>>>>>>>>>>>>>>>> exist
that can show there are any gaps in the body >>>>>>>>>>>>>>>>>>>>>>>>>> of human
knowledge that can be expressed in language >>>>>>>>>>>>>>>>>>>>>>>>>> that begins
with a consistent set of basic facts and only >>>>>>>>>>>>>>>>>>>>>>>>>> applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that >>>>>>>>>>>>>>>>>>>>> you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess >>>>>>>>>>>>>>>>>>>> of Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too >>>>>>>>>>>>>>>>>> inexpressive.
Natural numbers are one of the best understood topics >>>>>>>>>>>>>>>>> of mathematics.
Therefore restricting the assumptions to natural >>>>>>>>>>>>>>>>> numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims >>>>>>>>>>>>>>>>> and proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some >>>>>>>>>>>>>>> of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be >>>>>>>>>>>>>> prime, it uses prime numbers to number the symbols, and >>>>>>>>>>>>>> prime powers for the position.
And consequently Gödel's example of an undecidable formula >>>>>>>>>>>>> is more complex
than needed. Which doesn't really matter as the complex >>>>>>>>>>>>> details can be
hidden behind definitions.
...We are therefore confronted with a proposition which >>>>>>>>>>>> asserts its own unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia
Mathematica And Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is >>>>>>>>>> incorrect.
Gödel's G when boiled down to its barest essence is simply >>>>>>>>>> semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a >>>>>>>>> simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The >>>>>>> usual
meaning of what I said is that you don't disagree with what I
said in
my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its
own unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
You have never revealed any unsound detail of the proof, not even
whether it uses a false premise or an invalid inference.
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not
ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in >>>> different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
On 11/4/2025 3:46 AM, Mikko wrote:
On 2025-11-03 23:40:06 +0000, olcott said:
On 11/2/2025 6:41 AM, Mikko wrote:
On 2025-11-01 13:50:53 +0000, olcott said:
On 11/1/2025 4:10 AM, Mikko wrote:
On 2025-10-31 12:09:48 +0000, olcott said:
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:Incompleteness is a misconception nothing more.
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring essential aspects is
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese
this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human
knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive.
Natural numbers are one of the best understood topics of mathematics.
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted mess:
G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was important to
cover languages that have no symbol for ':=' or '⊬'. >>>>>>>>>>>>>>
But that is irrelevant to my preference for the numbering method.
His result is supposed to apply to any formal language >>>>>>>>>>>>> as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any >>>>>>>>>>>> symbols other than those of PA. A proof that requires other symbols
proves nothing about PA.
If he started with Minimal Type Theory and the understanding >>>>>>>>>>> that cycles in the evaluation sequence of expressions indicate >>>>>>>>>>> that they are semantically unsound we would have never heard >>>>>>>>>>> of Incompleteness.
Sooner or later someone would have found incompleteness anyway. >>>>>>>>>
No, it is a part of analytic knowledge. You have not shown any error >>>>>>>> in Gödel's and other proofs that every theory of natural numbers is >>>>>>>> incomplete. All you have shown is that you don't like the fact and >>>>>>>> some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if >>>>>>>> MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
Irrelevant to my comment that some parts of the theory are not yet shown.
Its exactly the same thing a FOL with additional operators
to make it much more expressive. For example it can directly
expression any HOL logic expression.
You mean it includes "tertium non datur" and "the principle of explosion"? >>>>
I give an example.
Thank's but the definition of the theory is still missing.
The theory is entirely expressed by the conventional
meaning of it operators and the syntax that I provided.
They don't express any theory. Shall we conclude that there is no theory?
In other words you are trying to get away with
saying that you have no idea what this means: A ↔ B ???
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring essential aspects is
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese
this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human
knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive.
Natural numbers are one of the best understood topics of mathematics.
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is simply semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual >>>>>>>> meaning of what I said is that you don't disagree with what I said in >>>>>>>> my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the >>>>>> complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its own >>> unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
You have never revealed any unsound detail of the proof, not even
whether it uses a false premise or an invalid inference.
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
On 2025-11-04 18:05:34 +0000, olcott said:
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:The complexity of Gödel's sentence is nuts.
On 10/30/2025 5:50 AM, Mikko wrote:I did not say nor mean "you did not disagree with anything". >>>>>>>>> The usual
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said: >>>>>>>>>>>>>>>
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:That's true, and I would prefer that rather than >>>>>>>>>>>>>>>>> Gödel's mapping. But
On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote:
On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring >>>>>>>>>>>>>>>>>>>>> essential aspects is
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human >>>>>>>>>>>>>>>>>>>>>>> knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts >>>>>>>>>>>>>>>>>>>>>>>>>>> and only applies semantic logical entailment >>>>>>>>>>>>>>>>>>>>>>>>>>> to those facts and can
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct >>>>>>>>>>>>>>>>>>>>>>>>>>>>> reasioning" some reasning that >>>>>>>>>>>>>>>>>>>>>>>>>>>>> actually is correct.
When we require that correct reasoning is >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nobody has verified all of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> saviour Jesus Christ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> at the same time supports the idea that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the conventiona wya to >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> from greenOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
cheese and the Moon is not made from >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> green cheese
this proves that Donald Trump is the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Lord and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
There is no counter-example that can >>>>>>>>>>>>>>>>>>>>>>>>>>>> possibly exist
that can show there are any gaps in the body >>>>>>>>>>>>>>>>>>>>>>>>>>>> of human
knowledge that can be expressed in language >>>>>>>>>>>>>>>>>>>>>>>>>>>> that begins
with a consistent set of basic facts and >>>>>>>>>>>>>>>>>>>>>>>>>>>> only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>>
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify >>>>>>>>>>>>>>>>>>>>>>> that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted >>>>>>>>>>>>>>>>>>>>>> mess of Gödel
numbers that people mistook his succinct analysis >>>>>>>>>>>>>>>>>>>>>> as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too >>>>>>>>>>>>>>>>>>>> inexpressive.
Natural numbers are one of the best understood topics >>>>>>>>>>>>>>>>>>> of mathematics.
Therefore restricting the assumptions to natural >>>>>>>>>>>>>>>>>>> numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims >>>>>>>>>>>>>>>>>>> and proofs as
fimite strings as thats how they are usually >>>>>>>>>>>>>>>>>>> expressed anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it >>>>>>>>>>>>>>>>>> just becomes the notational base for the numbers). >>>>>>>>>>>>>>>>>
the proof of Gödel's theorem is simpler if the number >>>>>>>>>>>>>>>>> of symbols in
the alphabet is prime even if that would mean that some >>>>>>>>>>>>>>>>> of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be >>>>>>>>>>>>>>>> prime, it uses prime numbers to number the symbols, and >>>>>>>>>>>>>>>> prime powers for the position.
And consequently Gödel's example of an undecidable >>>>>>>>>>>>>>> formula is more complex
than needed. Which doesn't really matter as the complex >>>>>>>>>>>>>>> details can be
hidden behind definitions.
...We are therefore confronted with a proposition which >>>>>>>>>>>>>> asserts its own unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))). >>>>>>>>>>>>>> false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia >>>>>>>>>>>>>> Mathematica And Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness >>>>>>>>>>>> is incorrect.
Gödel's G when boiled down to its barest essence is simply >>>>>>>>>>>> semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a >>>>>>>>>>> simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that. >>>>>>>>>
meaning of what I said is that you don't disagree with what I >>>>>>>>> said in
my previous message about the complexity of Gödel's sentence. >>>>>>>>
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the >>>>>>> complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its
own unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
You have never revealed any unsound detail of the proof, not even
whether it uses a false premise or an invalid inference.
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
No, they don't. That is very clearly expressed.
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not
ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it
up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do
with it anything that would stick in an infinte loop.
Usually sticking in an infinite loop is not desired or even perimitted
but there are exceptions. It all depends on the purpose of the program.
If the prupose is to compute all digits of pi then an infinte loop is
needed.
On 11/5/2025 4:46 AM, Mikko wrote:
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not >>>>>> ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in >>>>>> different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do
with it anything that would stick in an infinte loop.
That you do not know prolog very well is not a rebuttal.
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Usually sticking in an infinite loop is not desired or even perimitted
but there are exceptions. It all depends on the purpose of the program.
If the prupose is to compute all digits of pi then an infinte loop is
needed.
On 11/5/2025 4:49 AM, Mikko wrote:
On 2025-11-04 18:05:34 +0000, olcott said:
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:The complexity of Gödel's sentence is nuts.
On 10/30/2025 5:50 AM, Mikko wrote:I did not say nor mean "you did not disagree with anything". The usual
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said: >>>>>>>>>>>>>>>>
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring essential aspects is
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
There is no counter-example that can possibly existOn 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green
cheese and the Moon is not made from green cheese
this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
that can show there are any gaps in the body of human
knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive.
Natural numbers are one of the best understood topics of mathematics.
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result. >>>>>>>>>>>>>>>>>>>>
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))). >>>>>>>>>>>>>>> false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is simply semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that. >>>>>>>>>>
meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence. >>>>>>>>>
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the >>>>>>>> complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its own >>>>> unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
You have never revealed any unsound detail of the proof, not even
whether it uses a false premise or an invalid inference.
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
No, they don't. That is very clearly expressed.
They don't answer the question why and how is G unprovable in F
their answer is that there is some number that some how says so.
On 2025-11-05 11:55:33 +0000, olcott said:
On 11/5/2025 4:46 AM, Mikko wrote:
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not >>>>>>> ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it >>>>>>> up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do
with it anything that would stick in an infinte loop.
That you do not know prolog very well is not a rebuttal.
That you can lie is not a rebuttal of my rebuttal.
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Usually sticking in an infinite loop is not desired or even perimitted
but there are exceptions. It all depends on the purpose of the program.
If the prupose is to compute all digits of pi then an infinte loop is
needed.
If you think there is something wrong in what I said then point out
the wrong words and tell what should be said instead. Until that we
may reagard my words as right and correct.
On 2025-11-05 11:52:40 +0000, olcott said:
On 11/5/2025 4:49 AM, Mikko wrote:
On 2025-11-04 18:05:34 +0000, olcott said:
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:The complexity of Gödel's sentence is nuts.
On 10/30/2025 5:50 AM, Mikko wrote:I did not say nor mean "you did not disagree with anything". >>>>>>>>>>> The usual
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring >>>>>>>>>>>>>>>>>>>>>>> essential aspects is
On 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all >>>>>>>>>>>>>>>>>>>>>>>>> human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts >>>>>>>>>>>>>>>>>>>>>>>>>>>>> and only applies semantic logical >>>>>>>>>>>>>>>>>>>>>>>>>>>>> entailment to those facts and can >>>>>>>>>>>>>>>>>>>>>>>>>>>>> express all of human knowledge without any >>>>>>>>>>>>>>>>>>>>>>>>>>>>> gaps.
There is no counter-example that can >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> possibly existOn 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> reasioning" some reasning that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> actually is correct. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is only semantic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Principle ofOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> said:Right, and the empirical falct that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nobody has verified all of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> saviour Jesus Christ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> at the same time supports the idea that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the conventiona wya to >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Lord and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
that can show there are any gaps in the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> body of human
knowledge that can be expressed in >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language that begins >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify >>>>>>>>>>>>>>>>>>>>>>>>> that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse- >>>>>>>>>>>>>>>>>>>>>>>> engineers
the required architecture of the system. >>>>>>>>>>>>>>>>>>>>>>>> Wittgenstein
did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted >>>>>>>>>>>>>>>>>>>>>>>> mess of Gödel
numbers that people mistook his succinct >>>>>>>>>>>>>>>>>>>>>>>> analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too >>>>>>>>>>>>>>>>>>>>>> inexpressive.
Natural numbers are one of the best understood >>>>>>>>>>>>>>>>>>>>> topics of mathematics.
Therefore restricting the assumptions to natural >>>>>>>>>>>>>>>>>>>>> numbers provdes the
basis for the most interesting result. >>>>>>>>>>>>>>>>>>>>>
Of course one can ask whether one can prove a >>>>>>>>>>>>>>>>>>>>> similar theorem about
finite strings. It is much simpler to express >>>>>>>>>>>>>>>>>>>>> claims and proofs as
fimite strings as thats how they are usually >>>>>>>>>>>>>>>>>>>>> expressed anyway. But
that requires that one first constructs a theory of >>>>>>>>>>>>>>>>>>>>> finite strings.
But we can show a direct one-to-one mapping of >>>>>>>>>>>>>>>>>>>> finite strings in a finite alphabet to natural >>>>>>>>>>>>>>>>>>>> numbers (it just becomes the notational base for the >>>>>>>>>>>>>>>>>>>> numbers).
That's true, and I would prefer that rather than >>>>>>>>>>>>>>>>>>> Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number >>>>>>>>>>>>>>>>>>> of symbols in
the alphabet is prime even if that would mean that >>>>>>>>>>>>>>>>>>> some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to >>>>>>>>>>>>>>>>>> be prime, it uses prime numbers to number the symbols, >>>>>>>>>>>>>>>>>> and prime powers for the position.
And consequently Gödel's example of an undecidable >>>>>>>>>>>>>>>>> formula is more complex
than needed. Which doesn't really matter as the complex >>>>>>>>>>>>>>>>> details can be
hidden behind definitions.
...We are therefore confronted with a proposition which >>>>>>>>>>>>>>>> asserts its own unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory >>>>>>>>>>>>>>>> https://philarchive.org/archive/PETMTT-4v2 // YACC BNF >>>>>>>>>>>>>>>> of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))). >>>>>>>>>>>>>>>> false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia >>>>>>>>>>>>>>>> Mathematica And Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness >>>>>>>>>>>>>> is incorrect.
Gödel's G when boiled down to its barest essence is simply >>>>>>>>>>>>>> semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a >>>>>>>>>>>>> simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that. >>>>>>>>>>>
meaning of what I said is that you don't disagree with what I >>>>>>>>>>> said in
my previous message about the complexity of Gödel's sentence. >>>>>>>>>>
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about >>>>>>>>> the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts
its own unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
You have never revealed any unsound detail of the proof, not even
whether it uses a false premise or an invalid inference.
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
No, they don't. That is very clearly expressed.
They don't answer the question why and how is G unprovable in F
their answer is that there is some number that some how says so.
That no proof proves G is sufficient to say that G is unprovable.
On 11/6/2025 2:35 AM, Mikko wrote:
On 2025-11-05 11:55:33 +0000, olcott said:
On 11/5/2025 4:46 AM, Mikko wrote:
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not >>>>>>>> ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do
with it anything that would stick in an infinte loop.
The key idea is that the expression has a cycle in the
directed graph of its evaluation sequence. This does
make the expression semantically unsound.
On 11/6/2025 2:39 AM, Mikko wrote:
On 2025-11-05 11:52:40 +0000, olcott said:
On 11/5/2025 4:49 AM, Mikko wrote:
On 2025-11-04 18:05:34 +0000, olcott said:
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:The complexity of Gödel's sentence is nuts.
On 10/30/2025 5:50 AM, Mikko wrote:I did not say nor mean "you did not disagree with anything". The usual
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:As yousually your "in other words" is a lie. Existence of a simpler proof
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring essential aspects is
Categorically exhaustive reasoning reverse- engineers >>>>>>>>>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdfOn 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
There is no counter-example that can possibly existOn 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes thatOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green
cheese and the Moon is not made from green cheese
this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
that can show there are any gaps in the body of human
knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it. >>>>>>>>>>>>>>>>>>>>>>>>>
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive.
Natural numbers are one of the best understood topics of mathematics.
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result. >>>>>>>>>>>>>>>>>>>>>>
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory >>>>>>>>>>>>>>>>> https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))). >>>>>>>>>>>>>>>>> false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is simply semantically
unsound the same way the Liar Paradox is unsound. >>>>>>>>>>>>>>
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that. >>>>>>>>>>>>
meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence. >>>>>>>>>>>
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the >>>>>>>>>> complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound.
You have never revealed any unsound detail of the proof, not even
whether it uses a false premise or an invalid inference.
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
No, they don't. That is very clearly expressed.
They don't answer the question why and how is G unprovable in F
their answer is that there is some number that some how says so.
That no proof proves G is sufficient to say that G is unprovable.
If G is semantic gibberish then G would not be provable.
On 2025-11-06 11:03:30 +0000, olcott said:
On 11/6/2025 2:35 AM, Mikko wrote:
On 2025-11-05 11:55:33 +0000, olcott said:
On 11/5/2025 4:46 AM, Mikko wrote:
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not >>>>>>>>> ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed >>>>>>>>> it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do
with it anything that would stick in an infinte loop.
The key idea is that the expression has a cycle in the
directed graph of its evaluation sequence. This does
make the expression semantically unsound.
The expression "semanitcally unsound" does not mean anything if
no semantics is applied. It is also meaningless if the aplied
semantics does not assound any soundness to anything.
On 2025-11-06 11:08:22 +0000, olcott said:
On 11/6/2025 2:39 AM, Mikko wrote:
On 2025-11-05 11:52:40 +0000, olcott said:
On 11/5/2025 4:49 AM, Mikko wrote:
On 2025-11-04 18:05:34 +0000, olcott said:
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:You have never revealed any unsound detail of the proof, not even >>>>>>> whether it uses a false premise or an invalid inference.
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:The complexity of Gödel's sentence is nuts.
On 10/30/2025 5:50 AM, Mikko wrote:I did not say nor mean "you did not disagree with
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:As yousually your "in other words" is a lie. Existence of >>>>>>>>>>>>>>> a simpler proof
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>>
On 10/27/25 5:45 AM, Mikko wrote:And consequently Gödel's example of an undecidable >>>>>>>>>>>>>>>>>>> formula is more complex
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring >>>>>>>>>>>>>>>>>>>>>>>>> essential aspects is
Categorically exhaustive reasoning reverse- >>>>>>>>>>>>>>>>>>>>>>>>>> engineersOn 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all >>>>>>>>>>>>>>>>>>>>>>>>>>> human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> consistent set of basic facts >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and only applies semantic logical >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> entailment to those facts and can >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> express all of human knowledge without >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> any gaps.
There is no counter-example that can >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> possibly exist >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> body of humanOn 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> said:Only if you exclude from "correct >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> reasioning" some reasning that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> actually is correct. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is only semantic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the Principle of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
On 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Right, and the empirical falct that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nobody has verified all of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese
The conventional way to do proofs >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> from green >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Lord and savior Jesus Christ. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
3. Donald Trump is not the Lord and >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> saviour Jesus Christ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> at the same time supports the idea >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that the conventiona wya to >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
knowledge that can be expressed in >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language that begins >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to >>>>>>>>>>>>>>>>>>>>>>>>>>> verify that you have told >>>>>>>>>>>>>>>>>>>>>>>>>>> every necessary word to it. >>>>>>>>>>>>>>>>>>>>>>>>>>
the required architecture of the system. >>>>>>>>>>>>>>>>>>>>>>>>>> Wittgenstein
did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the >>>>>>>>>>>>>>>>>>>>>>>>>> convoluted mess of Gödel
numbers that people mistook his succinct >>>>>>>>>>>>>>>>>>>>>>>>>> analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously >>>>>>>>>>>>>>>>>>>>>>>> convoluted
way that Gödel added the functionality of this >>>>>>>>>>>>>>>>>>>>>>>> simple
operator: ⊢ to a formal language woefully too >>>>>>>>>>>>>>>>>>>>>>>> inexpressive.
Natural numbers are one of the best understood >>>>>>>>>>>>>>>>>>>>>>> topics of mathematics.
Therefore restricting the assumptions to natural >>>>>>>>>>>>>>>>>>>>>>> numbers provdes the
basis for the most interesting result. >>>>>>>>>>>>>>>>>>>>>>>
Of course one can ask whether one can prove a >>>>>>>>>>>>>>>>>>>>>>> similar theorem about
finite strings. It is much simpler to express >>>>>>>>>>>>>>>>>>>>>>> claims and proofs as
fimite strings as thats how they are usually >>>>>>>>>>>>>>>>>>>>>>> expressed anyway. But
that requires that one first constructs a theory >>>>>>>>>>>>>>>>>>>>>>> of finite strings.
But we can show a direct one-to-one mapping of >>>>>>>>>>>>>>>>>>>>>> finite strings in a finite alphabet to natural >>>>>>>>>>>>>>>>>>>>>> numbers (it just becomes the notational base for >>>>>>>>>>>>>>>>>>>>>> the numbers).
That's true, and I would prefer that rather than >>>>>>>>>>>>>>>>>>>>> Gödel's mapping. But
the proof of Gödel's theorem is simpler if the >>>>>>>>>>>>>>>>>>>>> number of symbols in
the alphabet is prime even if that would mean that >>>>>>>>>>>>>>>>>>>>> some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols >>>>>>>>>>>>>>>>>>>> to be prime, it uses prime numbers to number the >>>>>>>>>>>>>>>>>>>> symbols, and prime powers for the position. >>>>>>>>>>>>>>>>>>>
than needed. Which doesn't really matter as the >>>>>>>>>>>>>>>>>>> complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition >>>>>>>>>>>>>>>>>> which asserts its own unprovability. 15 ... >>>>>>>>>>>>>>>>>> (Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory >>>>>>>>>>>>>>>>>> https://philarchive.org/archive/PETMTT-4v2 // YACC BNF >>>>>>>>>>>>>>>>>> of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))). >>>>>>>>>>>>>>>>>> false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia >>>>>>>>>>>>>>>>>> Mathematica And Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931
Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is >>>>>>>>>>>>>>>> simply semantically
unsound the same way the Liar Paradox is unsound. >>>>>>>>>>>>>>>
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that. >>>>>>>>>>>>>
anything". The usual
meaning of what I said is that you don't disagree with what >>>>>>>>>>>>> I said in
my previous message about the complexity of Gödel's sentence. >>>>>>>>>>>>
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment >>>>>>>>>>> about the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts >>>>>>>> its own unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound. >>>>>>>
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
No, they don't. That is very clearly expressed.
They don't answer the question why and how is G unprovable in F
their answer is that there is some number that some how says so.
That no proof proves G is sufficient to say that G is unprovable.
If G is semantic gibberish then G would not be provable.
That does not follow.
G if provable if there is a proof that is
valid in the theory and ends with G. If there is not then G is
unprovable. Consequently, if G is not in the language of the
theory then it is unprovable.
But Gödel's G is a
sentence of the first order Peano arithmetic and
therefore has an arithmetic meaning. Gödel also proved that in its arithmetic meaning G is true. In some non-arthmetic meaning it can
be false.
On 11/7/2025 2:14 AM, Mikko wrote:
On 2025-11-06 11:03:30 +0000, olcott said:
On 11/6/2025 2:35 AM, Mikko wrote:
On 2025-11-05 11:55:33 +0000, olcott said:
On 11/5/2025 4:46 AM, Mikko wrote:
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not >>>>>>>>>> ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have mixed it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do >>>>>> with it anything that would stick in an infinte loop.
The key idea is that the expression has a cycle in the
directed graph of its evaluation sequence. This does
make the expression semantically unsound.
The expression "semanitcally unsound" does not mean anything if
no semantics is applied. It is also meaningless if the aplied
semantics does not assound any soundness to anything.
You are trying to be a s disagreeable as possible.
G := (F ⊬ G)
The above expression has a cycle in the directed
graph of its evaluation sequence making it incorrect.
On 11/7/2025 2:21 AM, Mikko wrote:
On 2025-11-06 11:08:22 +0000, olcott said:
On 11/6/2025 2:39 AM, Mikko wrote:
On 2025-11-05 11:52:40 +0000, olcott said:
On 11/5/2025 4:49 AM, Mikko wrote:
On 2025-11-04 18:05:34 +0000, olcott said:
On 11/4/2025 3:56 AM, Mikko wrote:
On 2025-11-03 23:45:57 +0000, olcott said:
On 11/2/2025 6:45 AM, Mikko wrote:You have never revealed any unsound detail of the proof, not even >>>>>>>> whether it uses a false premise or an invalid inference.
On 2025-11-01 13:54:09 +0000, olcott said:
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:The complexity of Gödel's sentence is nuts.
On 10/30/2025 5:50 AM, Mikko wrote:I did not say nor mean "you did not disagree with anything". The usual
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:As yousually your "in other words" is a lie. Existence of a simpler proof
On 2025-10-28 15:12:29 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>>>
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said: >>>>>>>>>>>>>>>>>>>>>>
On 10/26/25 7:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-25 19:57:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
On 10/25/2025 6:59 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-24 17:37:43 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
On 10/24/2025 2:46 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>Yes, an expectable consequence of ignoring essential aspects is
Categorically exhaustive reasoning reverse- engineersOn 10/23/2025 5:02 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Your words have much less material than all human knowledge without
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>>>>>>>>>>On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
There is no counter-example that can possibly existOn 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Only if you exclude from "correct reasioning" some reasning that
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said:
The conventional way to do proofs concludes that
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
within its rules if the Moon is made from green
cheese and the Moon is not made from green cheese
this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of
Explosion ceases to exist. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
actually is correct. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
that can show there are any gaps in the body of human
knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps.
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it. >>>>>>>>>>>>>>>>>>>>>>>>>>>
the required architecture of the system. Wittgenstein
did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple
operator: ⊢ to a formal language woefully too inexpressive.
Natural numbers are one of the best understood topics of mathematics.
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result. >>>>>>>>>>>>>>>>>>>>>>>>
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory >>>>>>>>>>>>>>>>>>> https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))). >>>>>>>>>>>>>>>>>>> false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is simply semantically
unsound the same way the Liar Paradox is unsound. >>>>>>>>>>>>>>>>
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that. >>>>>>>>>>>>>>
meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence. >>>>>>>>>>>>>
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
OK, you think Gödel's numbering scheme is simple.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ... (Gödel 1931:40-41)
It totally hides the details that proof his theorem is unsound. >>>>>>>>
The details of the proof that use Gödel numbers as their
primary basis hide the underlying semantics about exactly
what unprovable in F actually means.
No, they don't. That is very clearly expressed.
They don't answer the question why and how is G unprovable in F
their answer is that there is some number that some how says so.
That no proof proves G is sufficient to say that G is unprovable.
If G is semantic gibberish then G would not be provable.
That does not follow.
G := "lksef 34t987n m nk6f9-823t5jkn mhgtr,edgsbm"
try and prove that.
On 2025-11-07 13:00:44 +0000, olcott said:
On 11/7/2025 2:14 AM, Mikko wrote:
On 2025-11-06 11:03:30 +0000, olcott said:
On 11/6/2025 2:35 AM, Mikko wrote:
On 2025-11-05 11:55:33 +0000, olcott said:
On 11/5/2025 4:46 AM, Mikko wrote:
On 2025-11-04 17:45:04 +0000, olcott said:
On 11/4/2025 3:51 AM, Mikko wrote:
On 2025-11-04 00:32:51 +0000, olcott said:
On 11/3/2025 6:29 PM, Tristan Wibberley wrote:
On 03/11/2025 23:40, olcott wrote:
[the snip of the century]
G := (F ⊬ G)
G is defined as unprovable from F.
That is the ordinary "defined as" operator.
That's "definitionally reduces to", I think. If "(F ⊬ G)" is not >>>>>>>>>>> ultimately defined then G can still be not defined.
"define" /is/ a funny word isn't it? Several fields have >>>>>>>>>>> mixed it up in
different ways.
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
The point is that Prolog determines it is semantically
unsound because it has a cycle in the directed graph of
its evaluation sequence.
Prolog does not deremine it semantically unsound.
In other words you think that when the evaluation
of an expression gets stuck in an infinite loop
because of the self-reference in this expression
that this expression is semantically sound?
First of all, it does not stick in an infinite loop if you don't do >>>>>>> with it anything that would stick in an infinte loop.
The key idea is that the expression has a cycle in the
directed graph of its evaluation sequence. This does
make the expression semantically unsound.
The expression "semanitcally unsound" does not mean anything if
no semantics is applied. It is also meaningless if the aplied
semantics does not assound any soundness to anything.
You are trying to be a s disagreeable as possible.
A truth is a truth even when you disagree.
G := (F ⊬ G)
The above expression has a cycle in the directed
graph of its evaluation sequence making it incorrect.
By the usual rules it is syntactiaclly incorrect because he symbol on
the left side of := is used on the right side.
But that is irrelevant to Gödel's proof where G is not used before
nor in the definition of G.
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