Godel didn't hide steps regarding Godel numbering or diaganolization. Rather they are treated in exact detail.

"x is true if and only p" is not, according to Tarski or anyone who has reasonably studied this subject, the liar paradox nor the liar sentence. Moreover, as has been explained several times to the poster, Tarski does no use the liar sentence as a premise in any proof. Rather, Tarski assumes, toward a contradiction, that in the interpreted language there is a truth predicate for that language, and then shows that that assumption would allow the formation of the liar sentence and its contradiction, therefore that the assumption is contradictory and there is no such truth predicate.

"x is true if and only p" is not, according to Tarski or anyone who has reasonably studied this subject, the liar paradox nor the liar sentence. Moreover, as has been explained several times to the poster, Tarski does no use the liar sentence as a premise in any proof. Rather, Tarski assumes, toward a contradiction, that in the interpreted language there is a truth predicate for that language, and then shows that that assumption would allow the formation of the liar sentence and its contradiction, therefore that the assumption is contradictory and there is no such truth predicate. — TonesInDeepFreeze

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

My whole point in the post is that all epistemological antinomies such as the liar paradox
must be recognized and rejected thus not allowed to be any part of any undecidability proof.
It seems that you are saying that Tarski did not do that.

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems

It has been pointed out at least half a dozen times in other threads: Godel is referring to using the general idea of such paradoxes to spring the idea for his proof, but the actual proof does not use the liar paradox, as instead Godel constructs a sentence that says "I am unprovable" and not one that says "I am false". This can be verified by reading the actual proof.

And again, for about the tenth time: Tarski does not use the liar sentence as a premise in his proofs. Rather, for undefinability, he makes a reductio ad absurdum assumption that there is a truth predicate, from which he shows that that assumption provides a liar sentence that is a contradiction, thus refuting the reductio ad absurdum assumption. And in another proof, for incompleteness (undecidability actually), he explicitly says that he uses the predicate of provability not the predicate of truth. And one can verify in the actual proofs that the liar sentence is never a premise.

To put this in most stark form:

Assume P.
Derive Q.
Show that Q is contradictory.
Conclude ~P.

P is "a truth predicate can be formed"
Q is "the liar sentence can be formed"

The point is not that we adopt any liar sentence, but the opposite: that the liar sentence cannot be formed therefore there a truth predicate cannot be formed.

Tarski very much stresses that we do NOT deploy the liar sentence, since the liar sentence cannot even be formed in these kinds of languages.

To say that Tarski deploys the liar sentence in his proofs is to brazenly reverse what he wrote.

One more time: One can look at the actual proof steps for incompleteness and undefinability to see that the liar sentence is not used as a premise.

It has been pointed out at least half a dozen times in other threads: Godel is referring to using the general idea of such paradoxes to spring the idea for his proof, but the actual proof does not use the liar paradox, — TonesInDeepFreeze

That does not matter. That quote proved that he did not have the very basic understanding
that epistemological antinomies (AKA self-contradictory expressions) are not truth bearers
thus cannot be used for any undecidability proof. This showed that he generally had a poor
understanding of undecidability proofs.

According to my understanding of your explanation of Tarski, Tarski made an equivalent mistake.
He some how derived the Liar Paradox and did not reject it as not a truth bearer. Non-truth-bearers are a type mismatch error for any formal system of bivalent logic.

Saying again that Godel used the liar sentence in the incompleteness/undecidability proofs is to yet again ignore the plain hard fact that he did not.

Saying that Tarski derived the liar paradox is to yet again ignore the plain hard fact that he did not. Not only did Tarski not claim that the liar sentence is a truth bearer in the relevant formal systems, but even more fundamentally he showed the the liar sentence cannot even be formed in those formal systems. It really helps to actually study the subject of undefinability and incompleteness rather than to brazenly misrepresent it, as a seemingly perpetual strawman, as saying the opposite of what it actually says.

Tarski did not use the liar sentence as a premise, and he did not derive the liar sentence. Rather, he showed that a certain assumption would provide that the liar sentence could be formed in the interpreted language ((1) not even as a theorem, but merely as a sentence having the liar property, especially since there is no system of theorems involved but only an interpreted language, thus this is not a "derivation" of the liar sentence in the sense of a theorem, but only a proof that it could be formed as a sentence; moreover the derivation of the existence of the sentence is not a conclusion in the argument, but only a conditional result based on the reductio ad absurdum assumption) but if the liar sentence could be formed then it would be both true and false in the given interpretation, which contradicts that no sentence can be both true and false in an interpretation, thus we conclude the denial of the original assumption.

Saying again that Godel used the liar sentence in the incompleteness/undecidability proofs is to yet again ignore the plain hard fact that he did not. — TonesInDeepFreeze

I have not said anything like that. Here is what I said:

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

Saying that Tarski derived the liar paradox is to yet again ignore the plain hard fact that he did not. — TonesInDeepFreeze

It would then be possible to reconstruct the antinomy of the liar
in the metalanguage, by forming in the language itself a sentence x
such that the sentence of the metalanguage which is correlated
with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf

We can read the many posts in which the poster claimed that Godel used the liar sentence (i.e. the epistemological antinomy) in the proof.

And the quote of Godel just mentioned drops the context that Godel explicitly wrote this as a matter of "ANALOGY".

The wings of birds are analogous to the wings of airplanes, but they are not the same. The liar sentence "I am not true" is analogous to the Godel sentence "I am not provable", but they are not the same.

And the Tarski quote regards not the theorem that the poster previously cited (pages 275-276). Jumping around, changing contexts like that, is incoherent. But it goes right along with what I wrote. IF[/b] we had a truth predicate then we would have the liar paradox.

We can read the many posts in which the poster claimed that Godel used the liar sentence (i.e. the epistemological antinomy) in the proof. — TonesInDeepFreeze

Look is up in the proof yourself. Its one page 40

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:40)

One cannot correctly use epistemological antinomies in undecidability proofs,
they are not truth bearers and must be rejected on this basis.

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf

I am looking at the only Godel-approved translation right now. Page 598 of 'From Frege To Godel'.

"The ANALOGY of this argument [...]" [emphasis added ] which is the context of the footnote quoted.

Looking at the specific argument that Godel mentions as "this argument" we see that the liar sentence is not in that argument. Rather, Godel mentions that the liar sentence has an ANALOGOUS role.

So, again, I point out that the poster is quoting while intentionally omitting the crucial context.

Again, the wings of birds are analogous to the wings of airplanes, but they are not the same. The liar sentence "I am not true" is analogous to the Godel sentence "I am not provable", but they are not the same.

And again, if one actually reads the proof, then one will see that the liar sentence is not used anywhere in the proof.

Moreover, Godel could not use it as a line in any step of the proof, because the liar sentence cannot even be formulated in such systems that are the subject of the proof, which is what Tarski proved.

Moreover, Godel could not use it as a line in any step of the proof, because the liar sentence cannot even be formulated in such systems that are the subject of the proof, which is what Tarski proved. — TonesInDeepFreeze

Gödel is terribly wrong about this, these words are dead false:
His proof is an "undecidability proof" and he just proved that made a big mistake with
his understanding of undecidability proofs.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:40)
https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf

Godel is not wrong. What is wrong is brazenly, dishonestly attributing to him out of context, and then doing it yet again even after the context was reiterated yet again.

It's right there in the paper. Footnote 14 pertains to the passages that begin, "The analogy of this argument [...]"

It's right there in the paper. Footnote 14 pertains to the passages that begin, "The analogy of this argument [...]" — TonesInDeepFreeze

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:40)
https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf

There is no context in which those words of Gödel are not a terrible mistake.

The context in which they are not mistaken is the context in which he wrote them.

The poster seems to have a problem: Posting the quote from the footnote over and over again, as if the passage that it is a footnote to does not exist, even as the poster included my own quote of that passage.

The context in which they are not mistaken is the context in which he wrote them. — TonesInDeepFreeze

It seems that you are trying to take the words figuratively. That does not work.
Try and see how the literal meaning can be dismissed.

(1) The footnote pertains to the sense of using the antinomy analogously to a certain argument. That argument does not itself use the antinomy, but rather Godel's remarks show that the antinomy is analogous to that argument.

(2) That argument is then developed in full detail in a full proof. That proof does not use the liar sentence in any premise, line or conclusion. And the argument doesn't need to use the liar sentence in any premise, line or conclusion. Moreover, the liar sentence could not be used, since the liar sentence cannot even be formulated in the system under consideration.

(3) Textbooks that prove incompleteness do so without using the liar sentence as a premise, line or conclusion.

(4) Tarski even proved that the liar sentence cannot be formulated in such systems.

In all that context, it is seen that Godel did not use the liar sentence in the proof and did not say he did. What he said is that such antinomies can be use for such proofs, which in all this context, can only be understood not as used in the proofs formally but as an analogy that is adapted to the proofs. That adaptation is using 'provable' rather than 'true'. And that is also exactly what Tarski mentions explicitly in one of his proofs of incompleteness. Anyone with a sincere interest in understanding this subject sees this clearly, as opposed to someone who is interested only in taking a footnote out of context (to the extent of ignoring the very passage to which it is a footnote) for the purpose of insisting on a claim to be right no matter how ridiculous.

(4) Tarski even proved that the liar sentence cannot be formulated in such systems. — TonesInDeepFreeze

So lets get back to Tarski. He did anchor his proof in the Liar Paradox and he says so.
Try and show all of the details of otherwise.

I do not concede the prior point, but, let's move on.

In another thread, I showed exactly that Tarski did not use the liar sentence in the proof discussed there.

For the 100th time, Tarski himself said that instead of "true" he used "provable. It's in the exact text of the paper.

I guess the poster won't concede that footnote 14 is to the passages that begin by saying that the antinomies are analogous to the Godel argument.

Rational discussion is barely possible with a person who doen't have the honesty to recognize the connection between a footnote and what it footnotes.

For the 100th time, Tarski himself said that instead of "false" he used "unprovable. It's in the exact text of the paper. — TonesInDeepFreeze

He did start with this Liar paradox. He said so.
Also this is how he encoded his Liar Paradox
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x

https://liarparadox.org/Tarski_275_276.pdf
Actually the paper says he swapped "Tr" for "Pr"

I guess the poster won't concede that footnote 14 is to the passages that begin by saying that the antinomies are analogous to the Godel argument. — TonesInDeepFreeze

That is not the part I am not conceding. I am saying no matter
what anything says anywhere else THAT WAS A BIG MISTAKE

He said, even as you admit that he swapped, that the proof uses 'provable' instead of 'true'.

That is not the liar paradox.

It could only be a mistake if the footnote didn't pertain to the passages that begin with "The analogy [...]".

So if one concedes, by actually reading the paper, that the footnote pertains to those passages, then one doesn't have grounds to claim there is a mistake.

This stands:

https://thephilosophyforum.com/discussion/comment/907058

The proof uses the Godel sentence that is analogous to the liar sentence but is very different in crucial ways from the liar sentence. The proof itself does not have any mention of the liar sentence. The proof does not need the liar sentence. It would not even make sense for the proof to use the liar sentence. No one who has studied this subject thinks that the proof uses the liar sentence. The proof of just incompleteness doesn't even have to mention the notion of truth or semantics and can be formulated purely in finitisitc arithmetic and regarding only syntax.

Against all those facts, the only person who would claim that Godel is mistaken in this regard is someone obsessed with insisting that Godel is wrong no matter what, no matter how factually and logically wrong it is required to maintain that insistence.

In sum:

(1) In the Tarski proof of undecidability lately discussed here, Tarski did not use the liar sentence, but rather he used a different formulation involving provability rather than truth. And Tarski himself not only did not claim that the liar sentence has a truth value, but in the undefinability proofs he showed that in the relevant languages, it cannot even be formulated.

(2) In the Godel in the incompleteness paper, he did not claim that the liar sentence has a truth value, and he did not use the liar sentence in the actual incompleteness proof.

(1) In the Tarski proof of undecidability lately discussed here, Tarski did not use the liar sentence, — TonesInDeepFreeze

You seem to be consistently denying easily verified facts.

One thing that we do definitely do know about
Tarski's use of the "antinomy of the liar" in his Undefinability proof
is that he did not recognize it and reject it as a type mismatch error
for every formal system of bivalent logic.

It would then be possible to
reconstruct the antinomy of the liar in the metalanguage
by forming in the language itself a sentence x such that the
sentence of the metalanguage which is correlated with x
asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf — Tarski

Tarski's Undefinability Theorem Proof
https://liarparadox.org/Tarski_275_276.pdf

The easily verifiable fact is that in the undecidability proof, pages 275-276 of the paper, Tarski does not use the liar sentence in any step in that proof. Indeed, Tarski specifically mentions that he doesn't use the liar sentence but rather uses 'provable' rather than 'true'. This plain, incontrovertible fact has been brought to the poster's attention at least a dozen times.

And to evade that fact, the poster switches to a different theorem and proof elsewhere in the article, even though the context of that proof does not support the poster's false claim.

The poster just reposts over and over and over his same dogmatic and already rebutted out of context claims. At a certain point, replies are futile. The poster is out of reach of rational discussion.

The easily verifiable fact is that on page 275 of the undecidability proof the poster mentioned, Tarski does not use the liar sentence in any step in that proof. Indeed, Tarski specifically mentions that he doesn't use the liar sentence but rather uses 'provable' rather than 'true'. — TonesInDeepFreeze

Tarski's Liar Paradox from page 248
It would then be possible to reconstruct the antinomy of the liar
in the metalanguage, by forming in the language itself a sentence
x such that the sentence of the metalanguage which is correlated
with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf — Tarski

Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf

adapted to become this
x ∉ Pr if and only if p // line 1 of the proof

Here is (first three steps of) the Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p // assumption (see above)
(2) x ∈ True if and only if p // Tarski's convention T
(3) x ∉ Provable if and only if x ∈ True. // (1) and (2) combined

When L is Tarski's theory:
For the sentence "x is not provable in L" to be true in L requires a sequence of
inference steps in L that prove that they themselves do not exist.

First, as has been pointed out to the poster at least a dozen times, this is not a proof of undefinability. It is a proof of undecidability.

In that proof, Tarski does refer to the set of true sentences, but he does not use the liar sentence. Before the proof, Tarski specifically explains that he does not use the liar sentence, which pertains to truth, but rather a sentence regarding provability. And in the proof itself, he makes good on that by not using the liar sentence.

Notice also that the set Tr is defined in the meta-theory and not in the object theory.

And the poster repeats for the 10000th time his misconception that the theory would have to prove that the sentence is not provable. It's the opposite. The proof that the sentence is not provable does not occur in the theory but rather in the meta-theory, as the theory would be inconsistent if it proved that the sentence is not provable.

First, as has been pointed out to the poster at least a dozen times, this is not a proof of undefinability. It is a proof of undecidability. — TonesInDeepFreeze

The undefinability theorem is conventionally attributed to Alfred Tarski. Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1933 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann. Tarski had obtained almost all results of his 1933 monograph "The Concept of Truth in the Languages of the Deductive Sciences" between 1929 and 1931, and spoke about them to Polish audiences. However, as he emphasized in the paper, the undefinability theorem was the only result he did not obtain earlier. https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem — Wikipedia

has this as its original source:

from the original (PDF) on 9 January 2014. Retrieved 26 June 2013.
Tarski, A. (1983). "The Concept of Truth in Formalized Languages" (PDF). In Corcoran, J. (ed.). Logic, Semantics, Metamathematics. Translated by J. H. Woodger. Hackett. English translation of Tarski's 1936 article.
http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Tarski%20-%20The%20Concept%20of%20Truth%20in%20Formalized%20Languages.pdf — Wikipedia on Tarski's undefinability theorem

Which is the paper that I have been citing.

Let's FULLY address this one single point before we attempt to address any other points.

And again, for about the tenth time: Tarski does not use the liar sentence as a premise in his proofs. Rather, for undefinability, he makes a reductio ad absurdum assumption that there is a truth predicate, from which he shows that that assumption provides a liar sentence that is a contradiction, — TonesInDeepFreeze

In other words he does not understand that the derived liar sentence must be rejected as a non-truth-bearer thus a type mismatch error for any formal system of bivalent logic.

You never seem to directly address these exact words and seem to always form a rebuttal by talking around these exact words.

It has been fully addressed that the proof of undecidability on pages 275-276 does not use the liar sentence at any step.

/

I am not opining about a notion of a "type mismatch error" so I don't have to rebut it. I do not address the notion of "type mismatch error" without a crisp definition of it, and I wouldn't approach it without the poster at least first understanding his misconceptions about the actual proofs, which I have more than fully addressed:

(1) The liar sentence is not in any step of Godel's incompleteness proof and not in any step in Tarski's undecidability proof. Instead, the Godel sentence "I am not provable" is used, which is crucially different from the liar sentence "I am not true". The poster still does not grasp this, but instead he cites, like a disinformation bot, a Godel footnote out of context of the passage it footnotes, as in that passage Godel explicitly says that reference to the liar paradox is by analogy. The poster needs to stop talking around this.

(2) And the liar sentence itself is not a step, and especially not a premise, in Tarski's undefinability proofs, but rather the proofs start with the assumption, toward a contradiction, that the language can defined its own truth predicate, then shows that that would allow the language to form the liar sentence, which would yield a contradiction, so we conclude the denial of the assumption that the language can define its own truth predicate. The poster still refuses to understand this point that is at the very heart of undefinability. The poster needs to stop talking around this.

(3) The incompleteness proof does not at all entail that the theory itself would have to prove that the Godel sentence is not provable. The exact opposite: We prove that the theory itself does not prove that the Godel sentence is not provable. This has been explained to the poster probably at least 20 times in this forum. The poster needs to stop talking around this.

Also:

(4) In another thread, that the definition of 'antinomy' (especially in philosophy and logic) is not merely 'self-contradictory', which can be verified by looking at a number of dictionaries and articles. The poster still has not recognized this fact.