But Godel was speaking of a small finite collection of axioms, not an axiomatic system that continues to increase without end. At what point does one initiate the drawing of conclusions? Tacking on the axiom of choice took math into new dimensions, as did infinity axioms. BOAK seems bewildering rather than enlightening, imo.

The incompleteness theorem applies to formal theories, not to any set that includes natural language expressions. Moreover, the incompleteness theorem applies only to formal theories of a certain kind.

And subsequent discussion has been confused on other crucial points.

In the following, the context is formal theories.

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A theory is a set of sentences closed under provability.

A theory has a language. For any language L, there are models for L.

For any model M for a language L, every sentence in L is either true or false, and not both, in M.

A model M for the language L for a theory T is a model of T if and only if every theorem of T is true in M.

A theory T is consistent if and only if there is no sentence S in the language for T such that both S and the negation of S are theorems of T.

A theory T is inconsistent if and only if T is not consistent.

A theory T is consistent if and only if there is an M that is a model of T.

A theory T is inconsistent if and only if there is not M that is a model of T.

A theory T is complete if and only if, for every sentence S in the language L for T, either S is a theorem of T or the negation of S is a theorem of T.

A theory T is incomplete if and only if T is not complete.

The Godel-Rosser incompleteness theorem is: If a theory T is recursively axiomatizable, sufficiently arithmetic, and consistent, then T is incomplete.

A formula is valid if and only if it is satisfied by all models and assignments for the variables.

The Godel completeness theorem (not to be confused with the incompleteness theorem) is that if a formula P is valid then P is derivable in the pure first order predicate calculus.

The soundness theorem is that if a formula P is derivable in the pure first order predicate calculus then P is valid.

There is a particular model of PA (first order Peano arithmetic) that we call "the standard model of arithmetic". Since there are sentences in the language for PA such that neither the sentence nor its negation is a theorem of PA, there are sentences that are true in the standard model that are not theorems of PA. Moreover, the particular sentence G (that we call "the Godel sentence") is proven outside of PA to be true in the standard model.

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No recursively axiomatizable theory has an uncountable number of axioms. Theories with an uncountable number of axioms are not a consideration regarding the incompleteness theorem. Indeed, a language for a formal theory has only a countable number of symbols, thus only a countable number of expressions, thus only a countable number of formulas, thus only a countable number of sentences, thus there can be only a countable number of axioms in any axiomatization of the theory.

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For any model M, there is the theory T whose theorems are all and only the sentences true in M. It was Tarski who proved "the undefinability of truth" theorem, which says that the set of sentences true in the standard model for the language of arithmetic is not definable in the language of arithmetic. For example, there is no formula in the language for PA that is true of all and only the Godel numbers of sentences true in the standard model of arithmetic.

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The incompleteness theorem may be proved finitistically and contructivistically. While the proof is complicated, its methods are reducible to the basic reasoning of arithmetic by finite operations on single, discrete token objects. Thus, if the proof is considered dubious, then so is the basic reasoning of arithmetic on single, discrete token objects, which is to say that if the proof is considered dubious, then even the most basic arithmetic should be considered dubious.

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To reiterate regarding this thread:

(1) The incompleteness theorem does not pertain to any set that includes informal expressions. And the incompleteness theorem does not pertain to sets of expressions other than those of recursively axiomatized, sufficiently arithmetic, and consistent theories.

(2) Sentences are not true or false in theories or systems. Rather sentences are true or false in models.

(3) No recursively axiomatizable theory has an uncountable number of axioms.

But Godel was speaking of a small finite collection of axioms — jgill

Not a finite set of axioms, rather a countably infinite set of axioms. However, indeed, not an uncountable set of axioms as was incorrectly claimed by a poster earlier in this thread.

But Godel was speaking of a small finite collection of axioms, not an axiomatic system that continues to increase without end. At what point does one initiate the drawing of conclusions? Tacking on the axiom of choice took math into new dimensions, as did infinity axioms. BOAK seems bewildering rather than enlightening, imo. — jgill

The key change is the unprovable in BOAK simply means untrue in the BOAK, thus cannot means that BOAK is incomplete. BOAK is merely the actual body of general analytical knowledge as of now.

The incompleteness theorem applies to formal theories, — TonesInDeepFreeze

The first incompleteness theorem states that in any consistent formal system F
within which a certain amount of arithmetic can be carried out, there are statements
of the language of F which can neither be proved nor disproved in
https://plato.stanford.edu/entries/goedel-incompleteness/

The BOAK can perform every arithmetic and logical operations, thus qualifies for Gödel’s Incompleteness. Knowledge specified in natural language has been formalized using Montague Grammar.

For any model M for a language L, every sentence in L is either true or false, and not both, in M. — TonesInDeepFreeze

My system fully integrates the model directly within the formal system such that the system can perform deductive inference on the basis of semantics.

A theory T is complete if and only if, for every sentence S in the language L for T, either S is a theorem of T or the negation of S is a theorem of T. — TonesInDeepFreeze

I have changed this.
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
becomes
(¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
Incompleteness utterly ceases to exist.

True(L,x) ≡ (L ⊢ x)
False(L,x) ≡ (L ⊢ ¬x)

For any model M, there is the theory T whose theorems are all and only the sentences true in M. It was Tarski who proved "the undefinability of truth" theorem, which says that the set of sentences true in the standard model for the language of arithmetic is not definable in the language of arithmetic. — TonesInDeepFreeze

Thus step (3) of Tarski's proof is rejected as a false assumption:
https://liarparadox.org/Tarski_275_276.pdf
(3) x ∉ Provable if and only if x ∈ True. // derived from (1) and (2)

As I said, the incompleteness theorem pertains only to recursively axiomatizable theories. Montague grammar does not in and of itself make a general set of statements, even ones regarded as formalized by Montague grammar, a recursively axiomatizable theory. Understanding of the incompleteness theorem may be obtained by study of a textbook in mathematical logic.

Tarski's proof makes no false assumptions, no matter whatever incoherent ersatz pseudo formulations a crank on the Internet wishes to cook up.

Not a finite set of axioms, rather a countably infinite set of axioms. — TonesInDeepFreeze

I hoped you would chime in. :cool:

Tarski's proof makes no false assumptions, no matter whatever incoherent ersatz pseudo formulations a crank on the Internet wishes to cook up. — TonesInDeepFreeze

(3) x ∉ Provable if and only if x ∈ True. — PL Olcott

That is Tarski's line (3) that says that expression of language x is only true if x cannot be proven.

It seems that Tarski is saying: If Y is true and X follows from Y then X is not true.

grammar does not in and of itself make a general set of statements, even ones regarded as formalized by Montague grammar, a recursively axiomatizable theory. — TonesInDeepFreeze

I am not talking about Montague grammar per say. I am talking about some system like Montague Grammar that can explain every detail about human general knowledge such that a machine acquires a fully human level of understanding. The Cyc Project has been doing this for decades. It is known as a knowledge ontology inheritance hierarchy. Not words and a list of meanings. A knowledge tree having an enormous number of connections to and from each unique sense meaning of every word.

Tarski says no such thing as claimed two posts above.

And, for the third time: The incompleteness theorem pertains only to recursively axiomatizable theories.

If I don't comment to further replies, it's because meaningful discussion is not possible, and engagement is a waste of time, with a person who refuses to properly inform himself on the subject matter.

Tarski says no such thing as claimed two posts above. — TonesInDeepFreeze

Here is Tarski saying exactly that: "(3) x ∉ Pr if and only if x ∈ Tr"

The exact same thing is on the bottom of the first page of Tarski's actual text:
https://liarparadox.org/Tarski_275_276.pdf

Which means x is not an element of Provable if and only if x is an element of True.
I apologize for not making this 100% perfectly clear in my prior replies.

And, for the third time: The incompleteness theorem pertains only to recursively axiomatizable theories. — TonesInDeepFreeze

The body of all analytical knowledge includes every recursively axiomatizable theory as a subset.

For the fourth time: The incompleteness theorem pertains only to recursively axiomatizable theories.

A set of statements that includes a proper subset that is a recursive set is not thereby itself a recursive set. And it is howlingly ridiculous to cite some set of statements that includes all recursive axiomatizations, such such a set is patently inconsistent. Claims about "every recursively axiomatizable theory" in a larger theory then with implications for incompleteness is howlingly ridiculous ignorance.

But as a free speech hawk, I do applaud The Philosophy Forum for not ejecting such patently misinformational postings even though that restraint comes at the price of contributing that much further to the degradation of sincere discussion on such topics.

I wrote here an outline of a proof of Tarski's theorem. But it incorrectly glossed over crucial details. So I have deleted it here.

It is recommended that anyone truly interested in the subject may read a book in mathematical logic (for example, Enderton's 'A Mathematical Introduction To Logic'), which is the remedy for continual exposure on Internet forums to the bizarre misrepresentations of the subject by self-misinformed, confused and dogmatically prolific Internet cranks.

This is pages from his paper. I have taken them to be
the actual proof of the undefinability theorem.
https://liarparadox.org/Tarski_275_276.pdf
What do you think that they mean?

This sure seems to be talking about undefinability to me

From page 276
For every deductive science in which arithmetic is contained
it is possible to specify arithmetical notions which, so to speak,
belong intuitively to this science, but which cannot be defined
on the basis of this science.

The body of all analytical knowledge — PL Olcott

Unless you can describe this vague notion as it might appear in a computer program - that is to say a list with #1, #2, . . . - I can't get beyond it to the conclusions you draw. @TonesInDeepFreeze is recognized as a go-to source on these kinds of subjects.

Unless you can describe this vague notion as it might appear in a computer program - that is to say a list with #1, #2, . . . - I can't get beyond it to the conclusions you draw. TonesInDeepFreeze is recognized as a go-to source on these kinds of subjects. — jgill

He seems to be disagreeing with Tarski.

https://en.wikipedia.org/wiki/Ontology_(information_science) is how the knowledge is stored.
It is easiest to simply imagine that all the [general] things known to humans that can be written down in language have already been written down. Now we have the {body of analytic knowledge}.

It is easiest to simply imagine that all the [general] things known to humans that can be written down in language have already been written down. Now we have the {body of analytic knowledge}. — PL Olcott

So if listed, the listing might have to be refined as new knowledge is accrued. Still way to vague for me, but others may feel differently. I admire your tenacity on the subject.

So if listed, the listing might have to be refined as new knowledge is accrued. Still way to vague for me, but others may feel differently. I admire your tenacity on the subject. — jgill

It is simply the ordinary and common body of general knowledge known to mankind. It could be updated or static. Once the notions of {analytic truthmaker} is understood it is easy to see how this conquers Tarski Undefinability and Gödel's 1931 incompleteness. An expression of language either has an {analytic truthmaker} or it is untrue. Undecidable is no longer an option.