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### Tarski Undefinability Theorem Succinctly Refuted

Posted: Tue Apr 02, 2019 9:48 pm
Tarski proves that the Liar Paradox is true in his meta-theory 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 meta-theory.

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 ill-formed 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:

Tarski Undefinability Theorem Succinctly Refuted
https://www.researchgate.net/publicatio ... ly_Refuted

### Re: Tarski Undefinability Theorem Reexamined

Posted: Tue Apr 02, 2019 11:58 pm
PeteOlcott wrote: Tue Apr 02, 2019 9:48 pm Tarski proved that the Liar Paradox: G ↔ ~(F ⊢ G) is true in his
meta-theory 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
Do you have a point?

Is Tarski's 'proof' true?

### Re: Tarski Undefinability Theorem Reexamined

Posted: Wed Apr 03, 2019 12:25 am
A_Seagull wrote: Tue Apr 02, 2019 11:58 pm
PeteOlcott wrote: Tue Apr 02, 2019 9:48 pm Tarski proved that the Liar Paradox: G ↔ ~(F ⊢ G) is true in his
meta-theory 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
Do you have a point?

Is Tarski's 'proof' true?
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.

### Re: Tarski Undefinability Theorem Reexamined

Posted: Wed Apr 03, 2019 11:20 pm
PeteOlcott wrote: Wed Apr 03, 2019 12:25 am
A_Seagull wrote: Tue Apr 02, 2019 11:58 pm
PeteOlcott wrote: Tue Apr 02, 2019 9:48 pm Tarski proved that the Liar Paradox: G ↔ ~(F ⊢ G) is true in his
meta-theory 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
Do you have a point?

Is Tarski's 'proof' true?
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.
Doesn't seem simple at all.....

### Re: Tarski Undefinability Theorem Reexamined

Posted: Wed Apr 03, 2019 11:34 pm
A_Seagull wrote: Wed Apr 03, 2019 11:20 pm
PeteOlcott wrote: Wed Apr 03, 2019 12:25 am
A_Seagull wrote: Tue Apr 02, 2019 11:58 pm

Do you have a point?

Is Tarski's 'proof' true?
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.
Doesn't seem simple at all.....
In propositional logic it only depends on this axiom: S ↔ ~S

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 3:52 am
PeteOlcott wrote: Wed Apr 03, 2019 11:34 pm
A_Seagull wrote: Wed Apr 03, 2019 11:20 pm
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.
Doesn't seem simple at all.....
In propositional logic it only depends on this axiom: S ↔ ~S
And what does that axiom have to do with truth?

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 4:02 am
A_Seagull wrote: Thu Apr 04, 2019 3:52 am
PeteOlcott wrote: Wed Apr 03, 2019 11:34 pm
A_Seagull wrote: Wed Apr 03, 2019 11:20 pm

Doesn't seem simple at all.....
In propositional logic it only depends on this axiom: S ↔ ~S
And what does that axiom have to do with truth?
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

Posted: Thu Apr 04, 2019 4:05 am
PeteOlcott wrote: Thu Apr 04, 2019 4:02 am
A_Seagull wrote: Thu Apr 04, 2019 3:52 am
PeteOlcott wrote: Wed Apr 03, 2019 11:34 pm

In propositional logic it only depends on this axiom: S ↔ ~S
And what does that axiom have to do with truth?
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.
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

Posted: Thu Apr 04, 2019 4:25 am
A_Seagull wrote: Thu Apr 04, 2019 4:05 am
PeteOlcott wrote: Thu Apr 04, 2019 4:02 am
A_Seagull wrote: Thu Apr 04, 2019 3:52 am

And what does that axiom have to do with truth?
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.
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.
The English language defines all the human knowledge that can be expressed in English and it does this in English.

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 4:32 am
PeteOlcott wrote: Thu Apr 04, 2019 4:25 am
A_Seagull wrote: Thu Apr 04, 2019 4:05 am
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.
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.
The English language defines all the human knowledge that can be expressed in English and it does this in English.
The English language defines nothing. In any case what has that to do with truth?

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 4:47 am
A_Seagull wrote: Thu Apr 04, 2019 4:32 am
PeteOlcott wrote: Thu Apr 04, 2019 4:25 am
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.
The English language defines all the human knowledge that can be expressed in English and it does this in English.
The English language defines nothing. In any case what has that to do with truth?
If the English language actually defines nothing then you didn't just say that in English.

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 8:56 am
PeteOlcott wrote: Thu Apr 04, 2019 4:47 am If the English language actually defines nothing then you didn't just say that in English.
I said it in some language that you understood. Is it English?

That's a decision problem...

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 10:28 am
PeteOlcott wrote: Thu Apr 04, 2019 4:47 am
A_Seagull wrote: Thu Apr 04, 2019 4:32 am
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.
The English language defines nothing. In any case what has that to do with truth?
If the English language actually defines nothing then you didn't just say that in English.
Lol

### Re: Tarski Undefinability Theorem Reexamined

Posted: Thu Apr 04, 2019 4:06 pm
A_Seagull wrote: Thu Apr 04, 2019 10:28 am
PeteOlcott wrote: Thu Apr 04, 2019 4:47 am
A_Seagull wrote: Thu Apr 04, 2019 4:32 am

The English language defines nothing. In any case what has that to do with truth?
If the English language actually defines nothing then you didn't just say that in English.
Lol
Unless you are just playing games I would estimate that you may not have a deep enough
understanding of these things to provide any useful feedback.

### Re: Tarski Undefinability Theorem Succinctly Refuted

Posted: Thu Apr 04, 2019 5:46 pm
PeteOlcott 1 - A_Seagull 0
Popcorn anyone?
EB