From Newsgroup: comp.ai.philosophy

The Gödel number G has the arithmetic property of "being unprovable in
F" in exactly the same sense that the ASCII encoding of "I went to
Walmart to buy a carton of Breyer's ice cream" has the arithmetic
property of "going to the store to buy some food."
Which is to say: in no sense whatsoever.

Neither number has any such property. The number is just a number. It
has arithmetic properties — primality, divisibility, magnitude. It does
not have semantic properties. Semantic properties are not the kind of
thing numbers can have.
--
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).

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From Newsgroup: comp.ai.philosophy

Hi,

You wrote "Gödel number", and not "number" alone.
So in the context for "Gödel number" is not ordinary
arithmetic, like the context for "number".

But rather arithmetic plus an excoding of proofs.

Hope this Helps!

Bye

olcott schrieb:

The Gödel number G has the arithmetic property of "being unprovable in
F" in exactly the same sense that the ASCII encoding of "I went to
Walmart to buy a carton of Breyer's ice cream" has the arithmetic
property of "going to the store to buy some food."
Which is to say: in no sense whatsoever.

Neither number has any such property. The number is just a number. It
has arithmetic properties — primality, divisibility, magnitude. It does not have semantic properties. Semantic properties are not the kind of
thing numbers can have.

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From Newsgroup: comp.ai.philosophy

On 30/05/2026 02:11, olcott wrote:

The Gödel number G has the arithmetic property of "being unprovable in
F" in exactly the same sense that the ASCII encoding of "I went to
Walmart to buy a carton of Breyer's ice cream" has the arithmetic
property of "going to the store to buy some food."

The properties mentioned above are not arithmetic properties. It does
not even make sense to say that a number is unprovable. The conclusion
of a proof is always a sentence, never a number.

It would make sense to say that the sentence represented by G in
Gödels's (or some other numbering system) is not provable. But that
does not matter much because every number can represent any sentence
in some numbering system.

A numbering system that maps sentences to numbers cannot be specified
in arithmetics. It can be specified in a larget system that includes
both arithmetis and finite strings (as sentences are finite strings).
--
Mikko
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From Newsgroup: comp.ai.philosophy

On 5/29/2026 10:20 PM, Mild Shock wrote:

Hi,

You wrote "Gödel number", and not "number" alone.
So in the context for "Gödel number" is not ordinary
arithmetic, like the context for "number".

But rather arithmetic plus an excoding of proofs.

Syntactic encoding just like this is syntactic encoding
for grocery stores:
"I went to Walmart to buy a carton of Breyer's ice cream"

Hope this Helps!

Bye

olcott schrieb:

The Gödel number G has the arithmetic property of "being unprovable in
F" in exactly the same sense that the ASCII encoding of "I went to
Walmart to buy a carton of Breyer's ice cream" has the arithmetic
property of "going to the store to buy some food."
Which is to say: in no sense whatsoever.

Neither number has any such property. The number is just a number. It
has arithmetic properties — primality, divisibility, magnitude. It
does not have semantic properties. Semantic properties are not the
kind of thing numbers can have.

--
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).
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From Newsgroup: comp.ai.philosophy

But relative to a Gödel encoding {n},
not every number n is a Gödel number G.
Saying "not |- G", that sentences talk

about them selves is nothing new:

"Only the fool would take trouble to verify
that his sentence was composed of ten a’s,
three b’s, four c’s, four d’s, forty-six e’s,
sixteen f’s, four g’s, thirteen h’s, fifteen i’s,
two k’s, nine l’s, four m’s, twentyfive n’s,
twenty-four o’s, five p’s, sixteen r’s,
forty-one s’s, thirty-seven t’s, ten u’s,
eight v’s, eight w’s, four x’s, eleven y’s,
twenty-seven commas, twenty-three apostrophes,
seven hyphens, and, last but not least, a single !"

the above is wrong it has 28 commas. In some
sense its a strong Gödel sentence where we
not only have not |- G but also |- not G.

So Prolog negation as failure and classical
negation would agree, because 0 ≠ 1 is provable.

Mikko schrieb:

On 30/05/2026 02:11, olcott wrote:

The Gödel number G has the arithmetic property of "being unprovable in
F" in exactly the same sense that the ASCII encoding of "I went to
Walmart to buy a carton of Breyer's ice cream" has the arithmetic
property of "going to the store to buy some food."

The properties mentioned above are not arithmetic properties. It does
not even make sense to say that a number is unprovable. The conclusion
of a proof is always a sentence, never a number.

It would make sense to say that the sentence represented by G in
Gödels's (or some other numbering system) is not provable. But that
does not matter much because every number can represent any sentence
in some numbering system.

A numbering system that maps sentences to numbers cannot be specified
in arithmetics. It can be specified in a larget system that includes
both arithmetis and finite strings (as sentences are finite strings).

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From Newsgroup: comp.ai.philosophy

My bad, one should call it rather a weak
Gödel number. Counting characters inside
an encoded sequences seems to a primitive

recursive predication. There the theory
is complete. And complete theories have
not T |- A implies T |- not A.

But the real Gödel sentence operated
inside an incomplete Theory.

Mild Shock schrieb:

But relative to a Gödel encoding {n},
not every number n is a Gödel number G.
Saying "not |- G", that sentences talk

about them selves is nothing new:

"Only the fool would take trouble to verify
that his sentence was composed of ten a’s,
three b’s, four c’s, four d’s, forty-six e’s,
sixteen f’s, four g’s, thirteen h’s, fifteen i’s,
two k’s, nine l’s, four m’s, twentyfive n’s,
twenty-four o’s, five p’s, sixteen r’s,
forty-one s’s, thirty-seven t’s, ten u’s,
eight v’s, eight w’s, four x’s, eleven y’s,
twenty-seven commas, twenty-three apostrophes,
seven hyphens, and, last but not least, a single !"

the above is wrong it has 28 commas. In some
sense its a strong Gödel sentence where we
not only have not |- G but also |- not G.

So Prolog negation as failure and classical
negation would agree, because 0 ≠ 1 is provable.

Mikko schrieb:

On 30/05/2026 02:11, olcott wrote:

The Gödel number G has the arithmetic property of "being unprovable
in F" in exactly the same sense that the ASCII encoding of "I went to
Walmart to buy a carton of Breyer's ice cream" has the arithmetic
property of "going to the store to buy some food."

The properties mentioned above are not arithmetic properties. It does
not even make sense to say that a number is unprovable. The conclusion
of a proof is always a sentence, never a number.

It would make sense to say that the sentence represented by G in
Gödels's (or some other numbering system) is not provable. But that
does not matter much because every number can represent any sentence
in some numbering system.

A numbering system that maps sentences to numbers cannot be specified
in arithmetics. It can be specified in a larget system that includes
both arithmetis and finite strings (as sentences are finite strings).

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