Tarski Undefinability Theorem proof’s Error
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Tarski Undefinability Theorem proof’s Error
(1) x ∉ Provable if and only if p
(2) x ∈ True if and only if p
We shall show that the sentence x is actually undecidable and at the same time true.
(3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
we can derive the following theorems from the definition of truth
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
(6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable
(5) Provable(x) → True(x) // theorem
--refutes
(3) ~Provable(x) ↔ True(x) // assumption
Original Proof: https://liarparadox.org/Tarski_275_276.pdf
(2) x ∈ True if and only if p
We shall show that the sentence x is actually undecidable and at the same time true.
(3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
we can derive the following theorems from the definition of truth
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
(6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable
(5) Provable(x) → True(x) // theorem
--refutes
(3) ~Provable(x) ↔ True(x) // assumption
Original Proof: https://liarparadox.org/Tarski_275_276.pdf
Last edited by PeteOlcott on Thu Mar 30, 2023 4:43 am, edited 1 time in total.
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Re: Tarski Undefinability Theorem proof’s Error
"Mr. Gerald. Mr. Gerald!"
"Here! Yes?"
"The committee acknowledges your submission of the proof for Gadzang's 8th theorem. That out of the way, it'll please you to know we went through your 1500 page proof using Erin"s holes and the Batzinger series. The proof is almost perfect Mr. Gerald. "Almost" is a painful world in the world of mathematics as it is every other world we've discovered so far. Do you have a copy of your proof Mr. Gerald?"
"Really, there's an error in my proof? How? It's impossible. I checked it myself a thousand times. I made it a point to show every step, even the part where 1 = 0. Surely you must be joking, Mr. Kamura."
"We'll see about that Mr. Gerald. Please turn to page 1358, third paragraph, 8th line. I'm certain that a man of your caliber can see the glaring error therein."
"What are you talking about? I don't see a mistake. Can you be more specific?"
"More specific Mr. Gerald? Come now, Mr. Gerald, don't let's play games. Do you recall the Ventar incident in Pareda, 15th July 1643?"
"Yes, yes, I do. What of it? I don't see the connection."
"Members of the committee, as you can see, the deer fails to connect the dots, is unable to suss out the logically necessary connection between the flock of fowls in Astan's farm and the mole on Ms. Carmela's cheek. Your proof is wrong Mr. Gerald."
"But ... but ... deer? flock of fowls? Ms. who?"
"The exit is to the left Mr. Gerald. We thank you for, what?, how many years did you say you worked on the proof? 15? 20?"
"25"
"Good day Mr. Gerald."
"Here! Yes?"
"The committee acknowledges your submission of the proof for Gadzang's 8th theorem. That out of the way, it'll please you to know we went through your 1500 page proof using Erin"s holes and the Batzinger series. The proof is almost perfect Mr. Gerald. "Almost" is a painful world in the world of mathematics as it is every other world we've discovered so far. Do you have a copy of your proof Mr. Gerald?"
"Really, there's an error in my proof? How? It's impossible. I checked it myself a thousand times. I made it a point to show every step, even the part where 1 = 0. Surely you must be joking, Mr. Kamura."
"We'll see about that Mr. Gerald. Please turn to page 1358, third paragraph, 8th line. I'm certain that a man of your caliber can see the glaring error therein."
"What are you talking about? I don't see a mistake. Can you be more specific?"
"More specific Mr. Gerald? Come now, Mr. Gerald, don't let's play games. Do you recall the Ventar incident in Pareda, 15th July 1643?"
"Yes, yes, I do. What of it? I don't see the connection."
"Members of the committee, as you can see, the deer fails to connect the dots, is unable to suss out the logically necessary connection between the flock of fowls in Astan's farm and the mole on Ms. Carmela's cheek. Your proof is wrong Mr. Gerald."
"But ... but ... deer? flock of fowls? Ms. who?"
"The exit is to the left Mr. Gerald. We thank you for, what?, how many years did you say you worked on the proof? 15? 20?"
"25"
"Good day Mr. Gerald."
Re: Tarski Undefinability Theorem proof’s Error
Pete, I think you are posting in the wrong forum. You need to find a forum where people understand real mathematics. This is a forum for dipsticks who think they understand mathematics.
Re: Tarski Undefinability Theorem proof’s Error
That's such a weird claim.PeteOlcott wrote: ↑Wed Mar 29, 2023 5:01 am We shall show that the sentence x is actually undecidable and at the same time true.
To show something (anything) is to provide a decision procedure for the conclusion. What's otherwise called a proof.
So you are going to provide a decision procedure (a proof!) for True(x) while also showing that x is undecidable?
You don't have a clue what you are talking about.
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Re: Tarski Undefinability Theorem proof’s Error
If you look on the link you will see that this is Tarski's proofSkepdick wrote: ↑Sun Apr 02, 2023 3:05 pmThat's such a weird claim.PeteOlcott wrote: ↑Wed Mar 29, 2023 5:01 am We shall show that the sentence x is actually undecidable and at the same time true.
To show something (anything) is to provide a decision procedure for the conclusion. What's otherwise called a proof.
So you are going to provide a decision procedure (a proof!) for True(x) while also showing that x is undecidable?
You don't have a clue what you are talking about.
Re: Tarski Undefinability Theorem proof’s Error
I don't really care.PeteOlcott wrote: ↑Sun Apr 02, 2023 3:46 pm If you look on the link you will see that this is Tarski's proof
It's not Tarski's claim - it's your claim that "We shall show that the sentence x is actually undecidable and at the same time true."
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Re: Tarski Undefinability Theorem proof’s Error
It is a verbatim quote from Tarki's proof, you really must pay much better attentionSkepdick wrote: ↑Sun Apr 02, 2023 3:51 pmI don't really care.PeteOlcott wrote: ↑Sun Apr 02, 2023 3:46 pm If you look on the link you will see that this is Tarski's proof
It's not Tarski's claim - it's your claim that "We shall show that the sentence x is actually undecidable and at the same time true."
Re: Tarski Undefinability Theorem proof’s Error
I am paying attention, idiot. The sentence which follows immediately after is "For this purpose we shall pass to a metatheory of higher order.".PeteOlcott wrote: ↑Sun Apr 02, 2023 4:04 pm It is a verbatim quote from Tarki's proof, you really must pay much better attention
So he's not demonstrating that IN the theory.
He's demonstrating it in a metatheory - a model.
He's demonstrating it OUTSIDE the theory. Which is why he also says this:
According to Thesis A we can construct, on the basis of the enriched metatheory, a correct definition of truth concerning all the sentences of the theory studied.
By leaving out that (really fucking important!) part you are making a different claim to the one made by Tarski.
The truth predicate might be undefinable in the theory, but that doesn't mean it's undefinable in the metatheory.
It would really help if you understood the difference between proof theory (which is syntactic) and model theory (which is semantic)
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Re: Tarski Undefinability Theorem proof’s Error
By leaving that out we can analyze what he is saying on the same uniform basis and realize that he is trying to prove that a self-contradictory sentence is true.Skepdick wrote: ↑Sun Apr 02, 2023 4:07 pmI am paying attention, idiot. The sentence which follows immediately after is "For this purpose we shall pass to a metatheory of higher order.".PeteOlcott wrote: ↑Sun Apr 02, 2023 4:04 pm It is a verbatim quote from Tarki's proof, you really must pay much better attention
So he's not demonstrating that IN the theory.
He's demonstrating it in a metatheory - a model.
He's demonstrating it OUTSIDE the theory. Which is why he also says this:
According to Thesis A we can construct, on the basis of the enriched metatheory, a correct definition of truth concerning all the sentences of the theory studied.
By leaving out that (really fucking important!) part you are making a different claim to the one made by Tarski.
The truth predicate might be undefinable in the theory, but that doesn't mean it's undefinable in the metatheory.
It would really help if you understood the difference between proof theory (which is syntactic) and model theory (which is semantic)
Changing the frame-of-reference to his meta-theory is essentially the same thing as the fallacy of equivocation error.
This sentences is not true: "This sentences is not true" is true in his meta-theory (outer sentence) and self-contradictory in his theory (inner sentence).
Re: Tarski Undefinability Theorem proof’s Error
By leaving that out you are analyzing something he is neither saying nor doing.PeteOlcott wrote: ↑Sun Apr 02, 2023 4:38 pm By leaving that out we can analyze what he is saying on the same uniform basis and realize that he is trying to prove that a self-contradictory sentence is true.
By leaving it out you are opposing a strawman of your own making.
It's only an equivocation if he didn't tell you he's switching perspectives. But he does tell you, so it's not an error.PeteOlcott wrote: ↑Sun Apr 02, 2023 4:38 pm Changing the frame-of-reference to his meta-theory is essentially the same thing as the fallacy of equivocation error.
This is so trivial to any computer scientist I really can't fathom what's confusing you about it!
Here's a truth-predicate defined in the meta-theory such that....
Code: Select all
In [1]: def true(x):
...: if eval(x) == True:
...: return False
...: else:
...: return True
...:
In [2]: true("1+1==5")
Out[2]: True
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Re: Tarski Undefinability Theorem proof’s Error
The whole theory / meta-theory is nonsense used to fool the gullible. Tarski uses that as a scam to make it look like he proved that a self-contradictory sentence is true.Skepdick wrote: ↑Sun Apr 02, 2023 8:46 pmBy leaving that out you are analyzing something he is neither saying nor doing.PeteOlcott wrote: ↑Sun Apr 02, 2023 4:38 pm By leaving that out we can analyze what he is saying on the same uniform basis and realize that he is trying to prove that a self-contradictory sentence is true.
By leaving it out you are opposing a strawman of your own making.
It's only an equivocation if he didn't tell you he's switching perspectives. But he does tell you, so it's not an error.PeteOlcott wrote: ↑Sun Apr 02, 2023 4:38 pm Changing the frame-of-reference to his meta-theory is essentially the same thing as the fallacy of equivocation error.
People that are not dumber than a box of rocks understand that self-contradictory sentences are never true. They can see the scam for what it is.
Re: Tarski Undefinability Theorem proof’s Error
Idiots will remain idiots.PeteOlcott wrote: ↑Sun Apr 02, 2023 8:53 pm The whole theory / meta-theory is nonsense used to fool the gullible. Tarski uses that as a scam to make it look like he proved that a self-contradictory sentence is true.
People that are not dumber than a box of rocks understand that self-contradictory sentences are never true. They can see the scam for what it is.
Prove that the sentence is "self-contradictory" in the theory.
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Re: Tarski Undefinability Theorem proof’s Error
https://liarparadox.org/Tarski_247_248.pdfSkepdick wrote: ↑Sun Apr 02, 2023 8:57 pmIdiots will remain idiots.PeteOlcott wrote: ↑Sun Apr 02, 2023 8:53 pm The whole theory / meta-theory is nonsense used to fool the gullible. Tarski uses that as a scam to make it look like he proved that a self-contradictory sentence is true.
People that are not dumber than a box of rocks understand that self-contradictory sentences are never true. They can see the scam for what it is.
Prove that the sentence is "self-contradictory" in the theory.
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.
If we drop all the meta-language stuff as misdirection then we are
simply left with the liar paradox expressed directly in his theory.
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
The above axiom simply refutes the following assumption
(3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
Re: Tarski Undefinability Theorem proof’s Error
Do you continue to misunderstand that P → -P is not the same as ~P → P ?PeteOlcott wrote: ↑Sun Apr 02, 2023 9:31 pm (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
The above axiom simply refutes the following assumption
(3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
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Re: Tarski Undefinability Theorem proof’s Error
When we understand the underlying semantics of provability is a sequenceSkepdick wrote: ↑Sun Apr 02, 2023 9:45 pmDo you continue to misunderstand that P → -P is not the same as ~P → P ?PeteOlcott wrote: ↑Sun Apr 02, 2023 9:31 pm (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
The above axiom simply refutes the following assumption
(3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
of inference steps deriving x then we know Provable(x) is not merely a
meaningless propositional variable.
When x is proved from a sequence of inference steps and this same set of
inference steps proves that x is true: Provable(x) → True(x) then True(x) and
~Provable(x) is impossible.