Contrary to a claim made in this thread (and made by the same poster several other times in this forum), it is not the case the Godel sentence requires that there is a sequence of inference steps that prove that they don't exist (as has been explained several other times in this forum). — TonesInDeepFreeze

...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

The above is a direct quote from the proof and the system referred to by the above quote certainly does require a sequence of inference steps that prove that they themselves do not exist. The actual proof itself hides all of its underlying semantics behind the purely symbolic manipulation of mathematical operations. Diagonalization shows THAT G is unprovable in F and hides WHY G is unprovable in F.

More generally, Godel's and Tarski's proofs do not have the defects claimed in this thread (and claimed by the same poster several other times in this forum). That can be verified by reading an introductory textbook on mathematical logic in which the groundwork and proofs of Godel-Rosser incompleteness and Tarski undefinability are provided. — TonesInDeepFreeze

Tarski's proof is directly anchored in the actual Liar Paradox itself.
Liar Paradox basis of proof: https://liarparadox.org/Tarski_247_248.pdf
The actual proof itself: https://liarparadox.org/Tarski_247_248.pdf

Most people can understand that: "This sentence is not true" cannot possibly
be true or false thus is not a truth bearer. Tarski did not seem to understand that
or he would not have used it as the basis of his proof.

The incompleteness theorem requires no notion or terminology 'True(L, x)' where L is a set of axioms or system.

Rather, using the above style of notation, we have:

True(M x) where M is a model and x is a sentence. Read as "x is true in M".

and

Theorem(L x) where L is a set of axioms and x is a sentence. Read as "x is a theorem from L".

And we prove about certain systems:

Theorem(L x) implies that for every model M, if True(M y) for every y in L, then True(M x). (This is the soundness theorem).

And we prove about certain axiom sets:

If True(M x) in every model M, then Theorem(L x).

And incompleteness proves that there are M and L such that:

(1) For every y such that Theorem(M y), we have True(M y); and (2) True(M x); but (3) it is not the case that Theorem(L x).

"Did you lie?" doesn't have a truth value, because it is not a declarative sentence. Indeed, interrogatory sentences do not appear as lines in proofs. — TonesInDeepFreeze

Your statement here sounds nonsense. Some questions can be for true or false. For example,"You lied, didn't you?" This means you lied, and it is true. It is also to mean you should be aware of the fact that you lied. — Corvus

Corvus was testing the boundaries of what is included and what is not included by using a rhetorical question as a pseudo statement.

Again, as has been explained several times in this forum:

G asserts that G is not provable in system P.

But P does not prove G, and P does not prove that it does not prove G.

/

Proofs don't "hide" things. From fully declared axioms and rules of inference, we may prove Godel-Rosser. We may prove versions that do not mention semantics. And we may prove versions that mention both syntax and semantics. This is all famous and understood by reading an introductory textbook in mathematical logic.

Regarding Tarski's undefinablity theorem, Tarski proved that in certain systems, there does not even exist such a sentence. Not only did Tarski not use such sentences as a basis, he actually proved that such sentences don't even exist in the relevant systems. To not understand that is to not understand what the theorem is even about.

/

I know the context in which interrogatory sentences were mentioned lately. But the matter of interrogatories has been brought into other posts in this forum as part of incorrect attempts to refute the theorems.

As to manipulation of symbols, the incompleteness theorem can be be done in mere primitive recursive arithmetic, so the assumptions and means of reasoning are well within the scope of the methods of finite arithmetical calculations.

Proofs don't "hide" things. From fully declared axioms and rules of inference, we may prove Godel-Rosser. We may prove versions that do not mention semantics. And we may prove versions that mention both syntax and semantics. This is all famous and understood by reading an introductory textbook in mathematical logic. — TonesInDeepFreeze

When G asserts its own unprovability in F the proof of G in F does require a sequence of inference steps in F that prove that they themselves do not exist. We at the meta-math level can see that there cannot possibly be such a proof of G in F thus we know that the assertion that G is unprovable in F is true.

That unhides the whole essence of Gödel's proof where we can see WHY G is unprovable in F not merely THAT G is unprovable in F.

Regarding Tarski's undefinablity theorem, Tarski proved that in certain systems, there does not even exist such a sentence. Not only did Tarski not use such sentences as a basis, he actually proved that such sentences don't even exist in the relevant systems. To not understand that is to not understand what the theorem is even about. — TonesInDeepFreeze

When (as in Prolog) True(L, x) means Provable(L, x) and
(as in Prolog) False(L, x) means Provable(L, ~x) then Tarski Undefinability theorem utterly fails.
Self-contradictory expressions are simply rejected as not bearers of truth.

There is no proof of G in F.

That's the point.

Too miss that point is to utterly not know what the theorem is about.

"Why" is not a technical term, more a heuristic matter, and could mean different things to different people. In the most bare sense, "Why is T a theorem?" is answered by showing the proof of T . But, heuristically, there is a massive amount of discussion in the literature giving insight into the theorem and its proof; and insight is given in Godel's own paper.

If we define 'true' as 'provable', then of course all bets are off regarding these theorems as they are stated. And if in baseball we define 'hit' as 'home run', then we would throw away all the baseball statistics books. Yeah, we know all that.

But to accommodate someone who insists that 'true' means 'provable', then we could simply say that wherever we have written the word 'true', it is to be replaced by 'gorue'. Then read all the proofs and discussions about them with that change. It matters not toward understanding the substance of them.

There is no proof of G in F.

That's the point.

Too miss that point is to utterly not know what the theorem is about. — TonesInDeepFreeze

That there is no proof of nonsense does not make any formal system incomplete unless
Incomplete(F) is a euphemism for Incorrect(G).

One may consult introductory textbooks in mathematics to see how we can prove undefinability from incompleteness or prove incompleteness from undefinability.

The definition of 'incomplete' is simple:

A theory T is incomplete if and only if there is a sentence S in the language for T such that neither S nor its negation are a theorem of T.

It is trivial to prove that there are incomplete theories, and not trivial, though pretty easy, is proving the soundness theorem that then trivially proves the incompleteness of certain theories. What is interesting about Godel-Rosser is that there are incomplete theories of a particular kind (consistent, recursively axiomatizable and arithmetically sufficient).

A theory T is incomplete if and only if there is a sentence S in the language for T such that neither S nor its negation are a theorem of T. — TonesInDeepFreeze

The problem with this definition is that it proves that mathematical systems are "incomplete" when they cannot prove or refute nonsense. Self-contradictory expressions are nonsense and cannot be proven or refuted only because they are nonsense.

These are stipulative definitions. Anyone may use different definitions. To accommodate someone who insists that we don't use a technically defined term the way we have defined it, we could say 'gincomplete' instead. It would still be about the same point: If a theory T is consistent, recursively axiomatizable and arithmetically sufficient, then there is a sentence S such that neither it nor its negation is provable in T.

No self-contradiction is provable in a consistent theory, irrespective of incompleteness.

No self-contradiction is provable in a consistent theory, irrespective of incompleteness. — TonesInDeepFreeze

Gödel specifically states that the inability to prove a self-contradictory expression
DOES MAKE THE FORMAL SYSTEM INCOMPLETE.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Using stipulative definition Incomplete(F) is simply a euphemism for Incorrect(G).
Incomplete does not retain any of its conventional meaning
incomplete: not having all the necessary or appropriate parts.

Using stipulative definition this same way we could say that the inability of a formal system to prove a self-contradictory expression makes this formal system "A big fat cow".

Godel never said any such nonsense that if a system proves a contradiction then the system is incomplete. Indeed, if a system proves a contradiction then the system is complete.

Moreover the Godel sentence is not a self-contradiction.

Again, 'incomplete' in this context is given a stipulative technical definition pertaining to mathematical logic. The use of 'incomplete' in mathematics is not claimed to pertain to all the other everyday meanings or other technical meanings in other fields of study. The nature of stipulative technical definitions is not even something that one should have to point out in a philosophy forum.

Godel never said any such nonsense that if a system proves a contradiction then the system is incomplete. Indeed, if a system proves a contradiction then the system is complete. — TonesInDeepFreeze

You misquoted me. An epistemological antinomy <is> a self-contradictory expression

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Does say that the inability to prove a SELF-contradictory expression
"can likewise be used for a similar undecidability proof..."

I didn't quote.

The proof itself does not mention 'epistemological antinomy'. Godel's footnote pertains to analogies of the proof, the proof itself does not invoke a notion of 'epistemological antimony'. Godel is talking about heuristic insight there, which is an analogy (not an identification) between certain informal antinomies and his mathematical proof. It is quite an error to grasp onto a footnote out of context while ignoring the actual hard mathematical proof.

Again, however one characterizes the Godel sentence, it is not a contradiction. Indeed it is a true sentence of arithmetic.

Again, however one characterizes the Godel sentence, it is not a contradiction. Indeed it is a true sentence of arithmetic. — TonesInDeepFreeze

This G is unprovable in F because this G is nonsense in F
That G is nonsense in F does not show that there is anything wrong with F
the issue is ALL G's fault.

When G asserts its own unprovability in F the proof of G in F does require a sequence of inference steps in F that prove that they themselves do not exist. We at the meta-math level can see that there cannot possibly be such a proof of G in F thus we know that the assertion that G is unprovable in F is true.

That unhides the whole essence of Gödel's proof where we can see WHY G is unprovable in F not merely THAT G is unprovable in F. — PL Olcott

Mathematical logic does not assign "fault". Fault though would be vital to assign if one were a judge in a traffic accident case.

The Godel sentence is not a contradiction and it is not nonsense. It is a statement of arithmetic. And G is true, and G is true if and only if G is not provable in a theory such as PA. That doesn't make G a contradiction nor nonsense. And the theory itself does not prove G (first incompleteness) and the theory itself does not prove that it does not prove G (second incompleteness).

This all can be understood by simply reading an introductory textbook in mathematical logic.

Mathematical logic does not assign "fault". Fault though would be vital to assign if one were a judge in a traffic accident case. — TonesInDeepFreeze

By saying that F is incomplete when the real issue is that G is incorrect the blame
for the unprovability of G is F it misallocated.

The Godel sentence is not a contradiction and it is not nonsense. — TonesInDeepFreeze

The Gödel sentence itself cannot possibly be directly understood because all of its actual semantics are completely hidden from view. Because of this we must use these quotes to have a glimpse into his underlying reasoning:

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

When G asserts its own unprovability in F the proof of G in F does require a sequence of inference steps in F that prove that they themselves do not exist. That unhides the whole essence of Gödel's proof where we can see WHY G is unprovable in F not merely THAT G is unprovable in F.

This G is unprovable in F because this G is nonsense and Gödel expressly states that this kind of nonsense "can likewise be used for a similar undecidability proof"

I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted.

I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted. — TonesInDeepFreeze

You seem to simply ignore the main points that prove my case.

...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

When I talk about that quote you simply change the subject to something else totally ignoring my analysis of it.

No important point has been ignored [by me]. It's the other way around.

I pointed out that the footnote pertains to informal heuristic analogy and is not part of the formal proof itself. That is not changing the subject. And a proper analysis of the proof is not advanced by taking a footnote that is part of the informal remarks about the proof out of context. A proper analysis is to address the actual formal proof. Moreover, since Godel's original paper, the theorem has been strengthened to Godel-Rosser, and the context has been sharpened by the subjects of recursion and model theory, and the notation has been modernized, and the proofs have been streamlined, and the whole subject has been given greater elucidation and presentation. And all of that is provided by many introductory textbooks in mathematical logic. Thus, a proper analysis of incompleteness begins with study of an introductory textbook in mathematical logic. It is sure that flitting among snippets on the Internet and mere cursory readings of even original sources, while skimming all of that not toward step by step mathematical understanding and verification of inferences, to the point of seizing upon footnotes out of context and not understood, is not the way to an understanding or analysis of the subject.

By the way, the only authorized and authoritative translation is the one in van Heijenoort's anthology.

No important point has been ignored. It's the other way around. — TonesInDeepFreeze

The important point is that it self-contradictory expressions are still considered valid proof of undecidability and this proves that the entire notion of undecidability that ALL mathematical incompleteness depends is totally bogus. When we understand and accept that then your repeatedly going back to utterly extraneous details to this point is simply deflection.

I cannot provide for progress in a conversation by repeating that I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted.

I cannot provide for progress in a conversation by repeating that I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted. — TonesInDeepFreeze

Try and show how it makes sense to base undecidability on self-contradictory expressions or acknowledge that you do not understand that Tarski Undefinability is anchored in the Liar Paradox and we can move forward. Simply changing the subject to something else blocks all actual progress.

The posts have come full circle, at least three times today. If any new points arise, I'll consider addressing them.