Three laws of logic apply to all propositions
¬(p ∧ ¬p) Law of non-contradiction
(p ∨ ¬p) Law of excluded middle
p = p Law of identity

Because of Quine's paper: https://www.ditext.com/quine/quine.html most philosophers have been confused into believing that there is no such thing as expressions of language that are {true on the basis of their meaning}.

The unique contribution I have made to this is that the semantic meaning of these expressions is always specified by other expressions. When we can derive x or ~x by applying truth preserving operations to a set of semantic meanings then this perfectly aligns with Wittgenstein's concise critique of Gödel: https://www.liarparadox.org/Wittgenstein.pdf

Unless P or ~P has been proved in Russell's system P has no truth value and thus cannot be a proposition according to the law of the excluded middle. Sometimes this "proof" requires an infinite sequence of steps.

Now that the poster cannot support his dogmatically ignorant and confused claim — TonesInDeepFreeze

The sentence <IS> derived from his Liar Paradox
and has the same sort of paradoxical result.

I am taking your last post as an indication that you are really not interested
in an honest dialogue.

I have no false assumptions in this context. But one of the many false assumptions of the poster is that "This sentence is not provable" is the liar sentence. — TonesInDeepFreeze

"This sentence is not provable"
is not provable because that would require a sequence
of inference steps that prove that they themselves do not exist.

I did err in my previous post by overlooking the negation sign. And I should have emphasized again, for the 100th time, that 'unprovable' and 'untrue' are profoundly different. — TonesInDeepFreeze

Only when you fail to understand that True(L,x) requires a sequence of truth preserving operations from basic facts that are other expressions of language stipulated to be true.

That you have false assumptions does not make me incorrect.

'x e T if and only if p' is not the liar sentence. — TonesInDeepFreeze

You are not paying close enough attention

x ∉ True if and only if p
x {is not a member of} True

x ∈ True if and only if p
x {is a member of} True

If the liar sentence appears as a line in the undecidability proof then one could point exactly to the line. — 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

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 — PL Olcott

The formalized liar Paradox is adapted to become line 1 of the proof.

Anyone can look at pages 275-276 to see that no step in that proof uses the liar sentence. — TonesInDeepFreeze

You must carefully study pages 247, 248, 275, and 276.
I have spent hundreds of hours on those four pages over the last several years.

From this basis it is easy to see that every single detail that I said in the prior post
is exactly and precisely a verified fact.

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.

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.

Regarding Tarski, the poster just quotes again and again and again out of context and ignores the context explained to him dozens of times. He will continue to do that. At a certain point, replies are futile. — TonesInDeepFreeze

I am not a very good communicator, maybe this is more clear

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

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 the (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

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.

(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

One can look in a dictionary or in books and articles to see that there is more to being an antinomy than merely being a self-contradiction, especially in the subjects of philosophy or logic. — TonesInDeepFreeze

In any case we can simply cut-to-the-chase and carefully examine every single subtle
detail of Tarski's use of the "antinomy of the liar" in his Undefinability proof.

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

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

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"

(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.

An antinomy is not just any self-contradiction. — TonesInDeepFreeze

I have never seen any use of the term {epistemological antinomy} that did not mean self-contradictory.

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.

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.

Whatever the case about compositional meaning, 'epistemology antinomy' does not mean merely 'self-contradictory'. — TonesInDeepFreeze

antinomy seems to mean that by itself.

The meaning of 'antinomy' or 'epistemological antinomy' is not just 'self-contradictory'. — TonesInDeepFreeze

My only source for the meaning was the compositional meaning of the two separate terms.

relating to the study of the nature, origin, and limits of human knowledge
https://www.merriam-webster.com/dictionary/epistemological

A self-contradictory phrase such as "There is no absolute truth" can be considered an antinomy
https://en.wikipedia.org/wiki/Antinomy

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

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

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

Go look up Prior's approach to the liar and see what you think. — fdrake

Therefore Since This statement is False is , in other words, This Statement is true and this statement is false, this no more than p=(-p), which is a contradiction, which is from ealier just not the case.
https://thephilosophyforum.com/discussion/8134/liar-paradox-the-three-laws-of-logic-are-intact

ON A FAMILY OF PARADOXES
https://pages.nyu.edu/dorr/hempel/PriorTranslation.pdf

I just read the overview that was posted here. Pretty succinct.

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.

"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

Self contradictory expressions - assuming that is an expression which entails a contradiction or are otherwise equivalent to A&~A - are truth bearers. They have the capacity to be true or false - as in, it would mean something for the statement to be true, and mean something for the statement to be false. They just happen to be false. — fdrake

OK then try to explain how this sentence is a truth bearer: "This sentence is not true."

And any time you want to prove me wrong your keyboard is within easy reach. — tim wood

I either have to explain it in technical terms that you don't understand or explain
it in plain English where too much important meanings slip through the cracks.

The key most important point that can be summed up using the technical terms
of philosophy is that {epistemological antinomies} are not {truth-bearers}.

For example, let your database, described above, be represented by, "The cat sleeps on the couch." You now test "finite" strings against your database, and those that pass are "true," those that don't, "false." And you can do that. But what has that to do with true and false, or knowledge of any kind, or anything undecidable? How does that show that Godel was mistaken or incoherent? Or that Tarski was confused? — tim wood

Any cycle in the directed graph of the evaluation sequence of any expression conclusively proves that this expression is not a truth-bearer thus must be rejected by any formal system of bivalent logic.

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

This is also true for any {epistemological antinomy} AKA self-contradictory expression. If X cannot possibly be evaluated to true or false (for whatever reason) then X must be rejected by any formal system of bivalent logic.

It makes it undecidable - it appears you do not know what the words you use mean - and certainly not "inherently incorrect." — tim wood

A truth-bearer is an expression of language X that can be possibly evaluated to a Boolean value.

What the logicians call an undecidable expression X the philosophers of logic correctly assess
as not truth-bearer X.

Here is a much more formal way of saying the same thing:

Back in 2019 I created a formal system that detects cycles in the evaluation of an expression:
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF

Initially it took any MTT expression and output the directed graph
of the evaluation sequence of this expression. The current system
only outputs the XML of the expression yet the directed graph can
still be derived manually.

The directed graph nodes numbers are on the left (00,01,02) and the nodes
that they transition to (directed graph edges) are on the right (01,02,00)
LP := ~True(L, LP)
00 Not 01
01 True 02, 00 // cycle detected
02 L

It turns out the Prolog can also detect cycles in the directed
graph of the evaluation sequence of an expression.

The SWI-Prolog implementation of unify_with_occurs_check/2 is cycle-safe
and only guards against creating cycles, not against cycles that may
already be present in one of the arguments.
https://www.swi-prolog.org/pldoc/man?predicate=unify_with_occurs_check/2

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

In other words Prolog has detected a cycle in the directed graph of the
evaluation sequence of the structure of the Liar Paradox. Experts seem
to think that Prolog is taking "not" and "true" as meaningless and is
only evaluating the structure of the expression.

From reading your citation, it appears you're looking for a truth-maker for φ and not finding one. And from the reading, it is not clear that it needs one; i.e., φ is a truth-bearer. The problem, of course, is that if φ is true, then it is not true, and if it is not true, then it is true. — tim wood

Hence conclusively proving that φ cannot bear the truth value of true or the truth value of false.
That does not make True(L, φ) inconsistent. When True(L, φ) is false and True(L, ~φ) is false then φ is rejected are inherently incorrect. No sense moving beyond this point until after you totally get it.

From your Stanford reference we have, "Let φ be the sentence, "φ is not true." That is, we have φ, simple enough and we need nothing else.
-------------
Now, in plain English, using φ, describe how your "system" works. — tim wood

In plain English:
"φ is not true."
What is φ not true about?
φ is not true about being not true.
What is φ not true about being not true about?
φ is not true about being not true about being not true...
Ok so φ NEVER gets to the actual point.

Ok, let's think about some of these expressions. And no, I haven't seen the actual Milne's paper, so I cannot say more when I have just the one link. (Is it free and obtainable by the net?) — ssu

Not every truth has a truthmaker II PETER MILNE
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=e5f9578348844874f5d2542dcaff8d481e016483

Truthmaker Maximalism defended GONZALO RODRIGUEZ-PEREYRA
https://philarchive.org/archive/RODTMD

The above shows just what the problem is when you "Cantor's diagonalization" or basically negative self reference.

So what your problem in using diagonalization? — ssu

When we can directly see the cycle in the directed graph of the evaluation
sequence of an expression (thus not an acyclic directed graph) then we can
see the expression is not provable because there is something wrong with it.

When we hide this behind Gödelization and diagonalization we can still see
that that expression X is unprovable yet lose the fact that X is unprovable
because there is something wrong with it.

Undecidability, Provability, True(L,x) and Tarski Undefinability are all inherently interrelated.

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

The key missing piece is that no one ever noticed that epistemological antinomies are simply not truth-bearers, thus must be rejected by any formal system of bivalent logic. Once rejected they cannot form any basis for any undecidability proof.

From your Stanford reference we have, "Let φ be the sentence, "φ is not true." That is, we have φ, simple enough and we need nothing else.
-------------
Now, in plain English, using φ, describe how your "system" works.

What you claim is that it knows what is true and what is not. I'm very skeptical, because of many arguments against this, and also not least because you have been singularly non-responsive through at least two threads and many posts. — tim wood

I want to mostly Gödel and focus on how a True(L,x) predicate would actually apply to the properly formalize Liar Paradox.
LP := ~True(L, LP)

<Tarski Undefinability>
We shall show that the sentence x is actually undecidable
and at the same time true ... (page 275)

the proof of
the sentence x given in the metatheory can automatically be
carried over into the theory itself: the sentence x which is
undecidable in the original theory becomes a decidable sentence
in the enriched theory.
https://liarparadox.org/Tarski_275_276.pdf (page 276)
</Tarski Undefinability>

When we stick with theory L we get the same results, thus no need for any metatheory
True(L, LP) is false
True(L, ~LP) is false
Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))

So what Tarski says is undecidable in his theory is actually not a truth-bearer in his theory.
What Tarski said is provable in his metatheory making it true in his theory is
~True(L, LP) is true in his theory because LP is not a truth-bearer in L.

what Gödel shows is that there's a true, but unprovable sentences. — ssu

He doesn't actually show that and if he didn't hide his work we could see that he doesn't really show that. He doesn't even claim that, yet what he does claim is a little incoherent. G is true in PA yet not provable in PA. The way that we know G is true is that G is provable in meta-math.

Yet Milne has the gist of this: the problem here is that there indeed are true, but unprovable truths. — ssu

I have studied these actual papers. Milne is actually saying that there are some expressions that we know are true yet have no way what-so-ever to know that they are true. If an expression utterly lacks any criterion measure showing that it is true then it remains untrue.

The thing that all of these writers currently lack is shown below:
TT := True(L, TT)
TM := Has-a-Truthmaker(L, TM)
LP := ~True(L, LP)

True(L, TT) is false
True(L, ~TT) is false

True(L, TM) is false
True(L, ~TM) is false

True(L, LP) is false
True(L, ~LP) is false

Copyright 2024 PL Olcott

HERE IS WHY NONE OF THEM ARE TRUTH-BEARERS
BEGIN:(Clocksin & Mellish 2003:254)
equal(X, X).
?- equal(foo(Y), Y).

that is, they will allow you to match a term against an uninstantiated
subterm of itself. In this example, foo(Y) is matched against Y, which
appears within it. As a result,Y will stand for foo(Y), which is
foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))),
and so on. So Y ends up standing for some kind of infinite structure.
END:(Clocksin & Mellish 2003:254)

Let's stipulate that your "verbal model..," which is practically impossible and I suspect theoretically impossible, exists. I assume you mean that finite stings will be input, and that in every case the output will be, correctly, a T or an F. How will it work? — tim wood

We can see that when we formalize the Liar Paradox correctly
LP := ~True(L, LP)

and not the clumsy way that Tarski formalized it :
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x

There is no need for any separate theory and meta-theory.
True(L, LP) is false
True(L, ~LP) is false

Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))
Truthbearer(L,LP) is false

and we get the same result as the Tarski meta-theory directly in L
~True(L, LP) is true.

Gödel isn't just coming up with the Liar paradox and "hiding" the missing inference steps behind Gödel numbers and diagonalization. Many people do think that Gödel has fallen into the trap of self reference and is talking about basically the paradox, but he isn't. — ssu

Did you notice that I changed the subject to Tarski?
Tarski does the same thing as Gödel yet shows his work.

Does it escape your notice that the theory, to be efficacious in the desired manner, has been enriched and has to be enriched? Or that the enriched thing - or any enriched thing - is not the same as the thing not enriched? — tim wood

There has never actually been any need for this enrichment, it has always been
expressible in a single formal system with a single formal language as I elaborate below.

x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x

His intention was for formalize the actual Liar Paradox
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

The above is Is better stated as: p ↔ p ∉ True
his self-reference is a little clumsy and the above is still not quite
actual self-reference.

It is standard convention to formalize self-reference incorrectly
ϕ(x) there is a sentence ψ such that S ⊢ ψ ↔ ϕ⟨ψ⟩.
*The sentence ψ is of course not self-referential in a strict sense*,
but mathematically it behaves like one.
https://plato.stanford.edu/entries/self-reference/#ConSemPar
That whole article was ONLY about self-reference

x := y means x is defined to be another name for y
https://en.wikipedia.org/wiki/List_of_logic_symbols

Thus Tarski's p is better stated as: p := p ∉ True
even better as: p := ~True(Tarski-theory, p)
p says that it is not true in Tarski's-theory.

It took me 500 hours studying those four pages of Tarski to get to that point.
You probably won't be able to get there by skimming those two pages.

Then what did Godel do and how did he do it? Or rather, inasmuch as he rigorously derives his undecidable proposition, on what basis do you claim it impossible? And now I insist on your using English unless you are using symbols to prove/demonstrate a point. — tim wood

It is much simpler to see what Tarski did, Gödel hid the missing inference steps
behind Gödel numbers and diagonalization.

This is Tarski's formalized 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

This is stated more simply as LP := ~True(L ,LP)

Tarski found out that ~True(L, LP) is true (in his meta theory) and
True(L,LP) is not provable in his theory and this got him confused.

This sentence is not true: "This sentence is not true" is true because
"This sentence is not true" is not true.

https://liarparadox.org/Tarski_275_276.pdf