## Tarski Undefinability Theorem Succinctly Refuted

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PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Tarski Undefinability Theorem Succinctly Refuted

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
Last edited by PeteOlcott on Sun Apr 07, 2019 1:22 pm, edited 20 times in total.
A_Seagull
Posts: 895
Joined: Thu Jun 05, 2014 11:09 pm

### Re: Tarski Undefinability Theorem Reexamined

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?
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Tarski Undefinability Theorem Reexamined

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.
A_Seagull
Posts: 895
Joined: Thu Jun 05, 2014 11:09 pm

### Re: Tarski Undefinability Theorem Reexamined

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.....
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Tarski Undefinability Theorem Reexamined

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
A_Seagull
Posts: 895
Joined: Thu Jun 05, 2014 11:09 pm

### Re: Tarski Undefinability Theorem Reexamined

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?
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Tarski Undefinability Theorem Reexamined

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.
A_Seagull
Posts: 895
Joined: Thu Jun 05, 2014 11:09 pm

### Re: Tarski Undefinability Theorem Reexamined

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.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Tarski Undefinability Theorem Reexamined

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.
A_Seagull
Posts: 895
Joined: Thu Jun 05, 2014 11:09 pm

### Re: Tarski Undefinability Theorem Reexamined

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?
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Tarski Undefinability Theorem Reexamined

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.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

### Re: Tarski Undefinability Theorem Reexamined

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...
A_Seagull
Posts: 895
Joined: Thu Jun 05, 2014 11:09 pm

### Re: Tarski Undefinability Theorem Reexamined

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
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Tarski Undefinability Theorem Reexamined

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.
Speakpigeon
Posts: 976
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

### Re: Tarski Undefinability Theorem Succinctly Refuted

PeteOlcott 1 - A_Seagull 0
Popcorn anyone?
EB