Tarski Undefinability Theorem Succinctly Refuted

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Tarski Undefinability Theorem Succinctly Refuted
Tarski proves that the Liar Paradox is true in his metatheory and not provable in his theory. By creating three universal truth predicates that Tarski presumed could not possibly exist I prove that the Liar Paradox is false in his theory with no need to reference any metatheory.
The key aspect of my proof is that I provide axiom of Truth (3) that correctly decides that some expressions of language such as the formalized Liar Paradox are either illformed or false. We evaluate these as not true.
Truth Predicate Axioms
(1) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
(2) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (False(F, x) ↔ (F ⊢ ~x))
(3) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (~True(F, x) ↔ ~(F ⊢ x))
Formalizing the Liar Paradox in this way:
True(F, G) ↔ ~(F ⊢ G)
it becomes equivalent to Tarski’s third equation:
3) x ∉ Pr ↔ x ∈ Tr
By Truth axiom (3) we substitute ~True(F, G) for ~(F ⊢ G)
deriving True(F, G) ↔ ~True(F, G) ∴ the Liar_Paradox is false in F.
This causes the Tarski Proof to fail at his third equation.
The above can only be properly understood within the context
of the following four pages of the Tarski Paper:
http://liarparadox.org/247_248.pdf
http://liarparadox.org/Tarski_Proof_275_276.pdf
Tarski Undefinability Theorem Succinctly Refuted
https://www.researchgate.net/publicatio ... ly_Refuted
The key aspect of my proof is that I provide axiom of Truth (3) that correctly decides that some expressions of language such as the formalized Liar Paradox are either illformed or false. We evaluate these as not true.
Truth Predicate Axioms
(1) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
(2) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (False(F, x) ↔ (F ⊢ ~x))
(3) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (~True(F, x) ↔ ~(F ⊢ x))
Formalizing the Liar Paradox in this way:
True(F, G) ↔ ~(F ⊢ G)
it becomes equivalent to Tarski’s third equation:
3) x ∉ Pr ↔ x ∈ Tr
By Truth axiom (3) we substitute ~True(F, G) for ~(F ⊢ G)
deriving True(F, G) ↔ ~True(F, G) ∴ the Liar_Paradox is false in F.
This causes the Tarski Proof to fail at his third equation.
The above can only be properly understood within the context
of the following four pages of the Tarski Paper:
http://liarparadox.org/247_248.pdf
http://liarparadox.org/Tarski_Proof_275_276.pdf
Tarski Undefinability Theorem Succinctly Refuted
https://www.researchgate.net/publicatio ... ly_Refuted
Last edited by PeteOlcott on Sun Apr 07, 2019 1:22 pm, edited 20 times in total.
Re: Tarski Undefinability Theorem Reexamined
Do you have a point?PeteOlcott wrote: ↑Tue Apr 02, 2019 9:48 pmTarski proved that the Liar Paradox: G ↔ ~(F ⊢ G) is true in his
metatheory and not provable in his theory without ever realizing
that the only reason it is not provable in his theory is that it is not
true in his theory.
https://www.researchgate.net/publicatio ... Reexamined
Is Tarski's 'proof' true?

 Posts: 734
 Joined: Mon Jul 25, 2016 6:55 pm
Re: Tarski Undefinability Theorem Reexamined
Tarski's proof is incorrect.A_Seagull wrote: ↑Tue Apr 02, 2019 11:58 pmDo you have a point?PeteOlcott wrote: ↑Tue Apr 02, 2019 9:48 pmTarski proved that the Liar Paradox: G ↔ ~(F ⊢ G) is true in his
metatheory and not provable in his theory without ever realizing
that the only reason it is not provable in his theory is that it is not
true in his theory.
https://www.researchgate.net/publicatio ... Reexamined
Is Tarski's 'proof' true?
I will put it is the simplest possible terms. Tarski thinks that he proved
that Truth cannot be formalized because no formalization of Truth can
prove that a lie is True.
Re: Tarski Undefinability Theorem Reexamined
Doesn't seem simple at all.....PeteOlcott wrote: ↑Wed Apr 03, 2019 12:25 amTarski's proof is incorrect.A_Seagull wrote: ↑Tue Apr 02, 2019 11:58 pmDo you have a point?PeteOlcott wrote: ↑Tue Apr 02, 2019 9:48 pmTarski proved that the Liar Paradox: G ↔ ~(F ⊢ G) is true in his
metatheory and not provable in his theory without ever realizing
that the only reason it is not provable in his theory is that it is not
true in his theory.
https://www.researchgate.net/publicatio ... Reexamined
Is Tarski's 'proof' true?
I will put it is the simplest possible terms. Tarski thinks that he proved
that Truth cannot be formalized because no formalization of Truth can
prove that a lie is True.

 Posts: 734
 Joined: Mon Jul 25, 2016 6:55 pm
Re: Tarski Undefinability Theorem Reexamined
In propositional logic it only depends on this axiom: S ↔ ~SA_Seagull wrote: ↑Wed Apr 03, 2019 11:20 pmDoesn't seem simple at all.....PeteOlcott wrote: ↑Wed Apr 03, 2019 12:25 amTarski's proof is incorrect.
I will put it is the simplest possible terms. Tarski thinks that he proved
that Truth cannot be formalized because no formalization of Truth can
prove that a lie is True.
Re: Tarski Undefinability Theorem Reexamined
And what does that axiom have to do with truth?PeteOlcott wrote: ↑Wed Apr 03, 2019 11:34 pmIn propositional logic it only depends on this axiom: S ↔ ~SA_Seagull wrote: ↑Wed Apr 03, 2019 11:20 pmDoesn't seem simple at all.....PeteOlcott wrote: ↑Wed Apr 03, 2019 12:25 am
Tarski's proof is incorrect.
I will put it is the simplest possible terms. Tarski thinks that he proved
that Truth cannot be formalized because no formalization of Truth can
prove that a lie is True.

 Posts: 734
 Joined: Mon Jul 25, 2016 6:55 pm
Re: Tarski Undefinability Theorem Reexamined
It turns out that axioms are the ultimate foundation of Truth,
it is the ONLY way that we know that a dog is a kind of animal,
and that 5 > 3.
Re: Tarski Undefinability Theorem Reexamined
Well, there are axioms and there are axioms.PeteOlcott wrote: ↑Thu Apr 04, 2019 4:02 amIt turns out that axioms are the ultimate foundation of Truth,A_Seagull wrote: ↑Thu Apr 04, 2019 3:52 amAnd what does that axiom have to do with truth?PeteOlcott wrote: ↑Wed Apr 03, 2019 11:34 pm
In propositional logic it only depends on this axiom: S ↔ ~S
it is the ONLY way that we know that a dog is a kind of animal,
and that 5 > 3.
Axioms can only define a particular logical system. Any inferences made from those axioms can only be considered to be 'true' within that particular logical system. And it would be an incestuous logical system that made reference to its own truth.

 Posts: 734
 Joined: Mon Jul 25, 2016 6:55 pm
Re: Tarski Undefinability Theorem Reexamined
The English language defines all the human knowledge that can be expressed in English and it does this in English.A_Seagull wrote: ↑Thu Apr 04, 2019 4:05 amWell, there are axioms and there are axioms.PeteOlcott wrote: ↑Thu Apr 04, 2019 4:02 amIt turns out that axioms are the ultimate foundation of Truth,
it is the ONLY way that we know that a dog is a kind of animal,
and that 5 > 3.
Axioms can only define a particular logical system. Any inferences made from those axioms can only be considered to be 'true' within that particular logical system. And it would be an incestuous logical system that made reference to its own truth.
Re: Tarski Undefinability Theorem Reexamined
The English language defines nothing. In any case what has that to do with truth?PeteOlcott wrote: ↑Thu Apr 04, 2019 4:25 amThe English language defines all the human knowledge that can be expressed in English and it does this in English.A_Seagull wrote: ↑Thu Apr 04, 2019 4:05 amWell, there are axioms and there are axioms.PeteOlcott wrote: ↑Thu Apr 04, 2019 4:02 am
It turns out that axioms are the ultimate foundation of Truth,
it is the ONLY way that we know that a dog is a kind of animal,
and that 5 > 3.
Axioms can only define a particular logical system. Any inferences made from those axioms can only be considered to be 'true' within that particular logical system. And it would be an incestuous logical system that made reference to its own truth.

 Posts: 734
 Joined: Mon Jul 25, 2016 6:55 pm
Re: Tarski Undefinability Theorem Reexamined
If the English language actually defines nothing then you didn't just say that in English.A_Seagull wrote: ↑Thu Apr 04, 2019 4:32 amThe English language defines nothing. In any case what has that to do with truth?PeteOlcott wrote: ↑Thu Apr 04, 2019 4:25 amThe English language defines all the human knowledge that can be expressed in English and it does this in English.A_Seagull wrote: ↑Thu Apr 04, 2019 4:05 am
Well, there are axioms and there are axioms.
Axioms can only define a particular logical system. Any inferences made from those axioms can only be considered to be 'true' within that particular logical system. And it would be an incestuous logical system that made reference to its own truth.
Re: Tarski Undefinability Theorem Reexamined
I said it in some language that you understood. Is it English?PeteOlcott wrote: ↑Thu Apr 04, 2019 4:47 amIf the English language actually defines nothing then you didn't just say that in English.
That's a decision problem...
Re: Tarski Undefinability Theorem Reexamined
LolPeteOlcott wrote: ↑Thu Apr 04, 2019 4:47 amIf the English language actually defines nothing then you didn't just say that in English.A_Seagull wrote: ↑Thu Apr 04, 2019 4:32 amThe English language defines nothing. In any case what has that to do with truth?PeteOlcott wrote: ↑Thu Apr 04, 2019 4:25 am
The English language defines all the human knowledge that can be expressed in English and it does this in English.

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 Joined: Mon Jul 25, 2016 6:55 pm
Re: Tarski Undefinability Theorem Reexamined
Unless you are just playing games I would estimate that you may not have a deep enoughA_Seagull wrote: ↑Thu Apr 04, 2019 10:28 amLolPeteOlcott wrote: ↑Thu Apr 04, 2019 4:47 amIf the English language actually defines nothing then you didn't just say that in English.
understanding of these things to provide any useful feedback.
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Re: Tarski Undefinability Theorem Succinctly Refuted
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