On 10/11/2026 12:11 PM, Ross Finlayson wrote:
On 10/11/2026 10:27 AM, olcott wrote:
On 10/11/2026 11:36 AM, Ross Finlayson wrote:
On 10/11/2026 06:00 AM, olcott wrote:
On 10/11/2026 4:02 AM, Mikko wrote:
On 06/10/2026 15:32, olcott wrote:
On 10/6/2026 2:59 AM, Mikko wrote:
On 04/10/2026 16:22, olcott wrote:
On 10/4/2026 6:30 AM, Mikko wrote:
On 02/10/2026 14:55, olcott wrote:The Goldbach conjecture is meaningful yet its truth value will be >>>>>>>>> unknown unless and until it proof can be algorithmically
On 10/2/2026 4:37 AM, Mikko wrote:
On 01/10/2026 17:47, olcott wrote:It turns out that automated proof systems expose
On 10/1/2026 5:03 AM, Mikko wrote:
On 29/09/2026 19:31, olcott wrote:
On 9/29/2026 11:01 AM, Python wrote:
No, Pete. Calling an undecidable sentence "out-of-scope" >>>>>>>>>>>>>>>> does not make undecidability disappear.
For a formal theory T, "G is undecidable in T" simply means >>>>>>>>>>>>>>>>
T |-/- G and T |-/- ~G
That is a statement entirely about proofs. No model theory >>>>>>>>>>>>>>>> is involved.
And G is not "out-of-scope": it is a perfectly well-formed >>>>>>>>>>>>>>>> sentence of the language of T. What is out-of-scope, >>>>>>>>>>>>>>>> apparently, is merely anything for which your proposed >>>>>>>>>>>>>>>> semantics fails to supply an answer.
Within PTS any expression that is neither provable nor >>>>>>>>>>>>>>> refutable
never acquires any semantic meaning.
There is one more thing one needs to consider: there is no >>>>>>>>>>>>>> complete
method to find out whether a sentence or its negation can be >>>>>>>>>>>>>> proven.
Consequently, it is not possible to construct a system that >>>>>>>>>>>>>> rejects
all undecidable sentences.
If back-chained inference from expression X to axioms >>>>>>>>>>>>> has a loop it is rejected.
Irrelevant. A direct proof is either a forward chain from >>>>>>>>>>>> premises
to conclusion and an indirect proof is a similar chain from tne >>>>>>>>>>>> negation of the axioms and conclusion to a contradiction. >>>>>>>>>>>
the correct details the most clearly. They used
back-chained inference. In any case when a finite
canonical proof is impossible then the expression
never gets semantically grounded.
Existence of a proof and existence of a meaning are irrelevant to >>>>>>>>>
compressed.
It truth value will be known as soon as its proof has been
chekcked and
found good. The proof can be algorithmically compressed as soon as >>>>>>>> it is
found, even before the verification, though that is irrelevant to >>>>>>>> the
knowledge of its truth. Thought it is possible that it will be >>>>>>>> found to
be unprovable. But none of that affects its meaning and
meaningfulness.
What PTS says about its meaning is irrelevant.
A finite proof might not exist.
It may forever remain unknowable
thus outside of the domain of knowledge.
That is one possibility, as already noted above. But it is also
possibe that a finite proof exists and is knowable but remains
forever unknown.
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of general knowledge.
Anything unknown for any reason is not a member of the body of
knowledge.
However, these possibilities are irrelevant to the meaning of the
of the conjecture, which we do know in spite or our ignorance of
its provability and truth value.
It might be the case that its truth value is unknowable.
It even might be possible that we can't ever know whether its truth >>>>>>>> value is unknowable.
I restricted my system to "the body of general knowledge that >>>>>>>>> can be
expressed in language" to handle these edge cases.
Then your system needs be updated when something unknonw becames >>>>>>>> known.
It is updated the same way that Pluto ceased to be a planet.
It is not updated unless somebody updates it. Also note that the
process
that resulted to Pluto not being regarded as a planet was not simple >>>>>> and
easy.
If no human mind knows a thing then it is outside
the body of knowledge.
Being meaningful is not a limitation, nor is the ability to prove >>>>>>>> that the commutativity law is false.each other. For example, the sentence ∀x ∀y (x * y = y * x) is >>>>>>>>>The Commutative Law For Multiplication
https://en.wikiversity.org/wiki/Basic_Laws_of_Algebra
meaningful in many theories and provable in some of them and >>>>>>>>>> unporvable others.
That is merely a limitation of the individual formal systems. >>>>>>>>
It is true, so that any attempt at proving that
it is false is incorrect.
Meaningless expressions are outside the body of knowledge.
The commutativity law is not meaningless, whther provable or not.
My focus has always been on "the body of general knowledge
that can be expressed in language".
That should include the meanings of the commutativity law and its
negation nad that in some cases both are unprovable but meaningful. >>>>>>
a mathematical property where changing the order of
the operands does not change the final result.
The proof that feline cats are a type of 15 story
office building comes up empty because it is false.
I never really cared that system X is too weak to fully express Y. >>>>>>Then you should not post to this discussion as its topic is whether >>>>>> PA is too weak to fully express the proof of G.
On the Relation Between Heyting’s
and Gentzen’s Approaches to Meaning
Dag Prawitz (2013)
"Gentzen’s idea could be summarized by saying that
the meaning of a sentence is determined by what
counts as a canonical proof of it..."
https://doi.org/10.1007/978-3-319-22686-6_2
This is the way that
"true on the basis of meaning expressed in language"
has always worked in actual reality.
A weaker system has a larger scope of applications, which is often >>>>>>>> quite useful.
counter-factual. "Weaker" is stipulated to mean a reduced scope.
No, "stronger" is stipualted to mean a reduced scope, i.e., greater >>>>>> ability to exclude from the scope what is not wanted.
Irrelevant to your attempt to deceive with an ireelevancy.For example, it is not provable in a group
(where its negation isn't, either) nor in the quaternions (where >>>>>>>>>> its negation is provable) but it is meaningful in both.
For
"true on the basis of meaning expressed in language"
when an expression cannot possibly connect to the
expressions that give it meaning then it never gets
any meaning.
That is not the basis in the above examples.
*This is the domain of my proposed system*
"the body of general knowledge that can be expressed in language" >>>>>>>>
If all words are defined in terms of other words andThey are also meaningless if there are diretly or indirectly >>>>>>>>>> circular definitions.
there is a string of symbols that it never assigned
any relation to other strings then it remains meaningless. >>>>>>>>>>
Like the cycle in the directed graph of the evaluation sequence >>>>>>>>> of the Liar Paradox formalized in Prolog.
Quite unlike. Circular definitions don't define. A cyclic
evaluation
graph
Never terminates thus never derives PTS semantic meaning.
Irrelevant. A cyclic evaluation graph is not meaningless. A circular >>>>>> definition does not present a meaning.
On the Relation Between Heyting’s
and Gentzen’s Approaches to Meaning
Dag Prawitz (2013)
"Gentzen’s idea could be summarized by saying that
the meaning of a sentence is determined by what
counts as a canonical proof of it..."
https://doi.org/10.1007/978-3-319-22686-6_2
This is the way that
"true on the basis of meaning expressed in language"
has always worked in actual reality.
It's a formal fact that any declaration has at least two readings,
for examples, one where it's true and one where it's not, and for
a wider example, one where it's true for itself (the usual "tau-
tology")
and one where it's as for "cosmic complement" given
an actual account of formal identity (the less-than-usual "identity"). >>>>
So its true that 5 equals 6 has more than one
correct truth value?
Then, for examples, a declaration is all the questions it answers,
implicitly with all the other declarations (and all the other
questions).
Questions may stay the same temporally, declarations their answers
change, with the times.
What's "true" is what remains invariant under alternatives,
tests and examinations, all the space of questions, all the
space of declarations, sentences with meaning are re-readable.
So, accounts of meaning or _significance_ vis-a-vis something like
Russell's "isolation" and "significance", about the signification
of statements, makes for mechanics in informatics and accounts of
work, then for both what statements "prove" or what can be inferred
from them, _and_ what "prove" them.
Then, "the meaning" of a declaration about the relate-able,
talking to yourself, then the relay-able, convey-able to others,
makes for that there's always again _another_ layer of meaning,
about the utterances and the utterers, and the audience, _another_
layer, point being usual accounts about "Sowa" and so on are
navel-gazing of the monomaniac variety.
Too convoluted for yo top show that you understand
the exact meaning of my words.
Really, lots of people reject and refuse "Tarski-trivial" or
"Wittgenstein-tautological" when it comes to "true",
Tarski as trivial, Wittgenstein as tautological.
There was a time when there was a universal consensus
that the Earth was flat.
There's Sheffer and Chwistek, and Huntington's universals, ....
In Tarski's theory of a flat earth it's flat,
and Wittgenstein says it doesn't matter anyways,
due _a priori_ truths like "a strict inequality
exists in ordinals as modeled by finite natural
integers" that "5 < 6 and 5 =/= 6 is a fact"
still has, for example, that neither five wrongs
nor six wrongs is anyways right.
Wittgenstein is a liar whose great contribution
to liars is "don't break the spell", Tarski is
a tabulator who has run out of fingers.
The "invincible ignorance of inductive inference"
is readily dis-spelled by deductive reasoning.
The polaricentric and other accounts of deductive
reasoning about space and space-time in the universals
and wider Mach-ian and wider LaGrange-ian is a sort of
ultimate refutation of "Flat-Earth".
Statements and declaration and interrogatives and
question in our rich natural language readily make
for that it's a formal fact that while transliterations
of words already have a "number", which is readily enough
their letters in their alphabet the word, they also have
a number that is infinite.
Matters of motion and continuity and other accounts of
ultimate universals make it so your mis-reading of Goedel
and Prawitz and so on then the ostrich-heading is bull-shit.
Quit trying, or change your ways.
Wittgenstein says things like "you can't talk about what's
not given in my theory" then "no-one can understand what
you say in your theory",
these are logically insane and
marks of a liar and abuser of language.
Now, a lot of people find it liberating to latch onto something that Wittgenstein says as giving them free-will when it's perceived accounts
like "determinism" or "natural order" or for that matter "positivism"
or even "nominalism" or "nominalist fictionalism" get in the way,
yet nobody needs Wittgenstein's permission or lack thereof for anything.
Tarski is the type to try to make a model of a logical theory--
that you can only follow not change. He failed trying to prove
he could make one.
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