experchange > comp.theory

peteolcott (03-04-19, 05:18 AM)


The only logic that you need to know is that English sentences are
assigned to variable names as an abbreviation of the whole sentence.

The following are simple English equivalents of steps (1)(2)(3) of the
above quoted Tarski proof: (see technical footnote)

x = "dogs are cats" // Give x a specific semantic value

(1) p = "[dogs are cats] cannot be proven"

(2) p = "[dogs are cats] is true"

(3) "[dogs are cats] cannot be proven" = "[dogs are cats] is true"

Now it is very obvious (in simple English) exactly how Tarski screwed up.

If we give x a specific semantic value then the third step of the Tarski
proof is obviously incoherent. The error emerges at the second step of
the proof.

As a personal note to: Franz Gnaedinger, The Tarski proof is merely a
enormously simplified paraphrase of Gödel's 1931 incompleteness theorem.

Anyone responding from sci.logic please convert all of your words
to simple English so that the linguists will be able to participate.

Technical Footnote: (for logicians)
(1) x ? Pr ? p // p ? ~Provable(x)
Where the symbol ?p? represents the whole sentence x. (thus showing that Tarski uses "?" to mean "=")
(2) x ? Tr ? p // p ? True(x)
(3) x ? Pr ? x ? Tr // ~Provable(x) ? True(x)
peteolcott (03-04-19, 08:14 AM)
On 3/3/2019 11:03 PM, DKleinecke wrote:
> On Sunday, March 3, 2019 at 7:18:33 PM UTC-8, peteolcott wrote:
> ERROR - you just gave p a different value


Technical Footnote: (for logicians)
(1) x ∉ Pr ↔ p // p ↔ ~Provable(x)
(2) x ∈ Tr ↔ p // p ↔ True(x)

If you look closely you will see that is Tarski's error not mine:
When we plug this value: "dogs are cats" into Tarski's: x
We get p with two different values, Tarski's mistake, not mine.

He goes on to combine the erroneous (1) and (2) in his step (3)
(3) x ∉ Pr ↔ x ∈ Tr // ~Provable(x) ↔ True(x)
peteolcott (03-04-19, 08:18 AM)
On 3/3/2019 11:11 PM, DKleinecke wrote:
> On Sunday, March 3, 2019 at 7:18:33 PM UTC-8, peteolcott wrote:
> ERROR - that this a valid representation of anything Tarski
> ever said.


I just converted his formulas to English:
(1) p = "x cannot be proven"
(2) p = "x is true"
(3) "x cannot be proven" = "x is true"

and plugged: "dogs are cats" into x.
[..]
peteolcott (03-04-19, 08:29 AM)
On 3/4/2019 12:18 AM, peteolcott wrote:
> On 3/3/2019 11:11 PM, DKleinecke wrote:
> I just converted his formulas to English:
> (1) p = "x cannot be proven"
> (2) p = "x is true"
> (3) "x cannot be proven" = "x is true"


Here is the original source:

I paid $35 for a hard copy just so that I could quote those two pages.
peteolcott (03-04-19, 04:46 PM)
On 3/4/2019 2:03 AM, Franz Gnaedinger wrote:
> On Monday, March 4, 2019 at 4:18:33 AM UTC+1, peteolcott wrote:
> Cats and dogs are mammals, both descending from a common ancestor,


If you pay any attention at all you will see that Tarksi's proof is
incorrect because it would show that because it cannot be proven
that {dogs are cats} this would make the sentence {dogs are cats} true.
[..]
peteolcott (03-04-19, 06:19 PM)
On 3/4/2019 9:13 AM, Rupert wrote:
> On Monday, March 4, 2019 at 4:04:37 PM UTC+1, peteolcott wrote:
> That's not the proof of the undefinability of truth. It seems to be an explanation of how to construct the Goedel sentence, which is a different thing..


This is the Tarski sentence: (3) x ? Pr ? x ? Tr
Which is an enormously simplified version of the Gödel sentence:


> If we look at the penultimate paragraph on p. 276, that may be a reference to the undefinability of truth, but the part that you are quoting is to do with the construction of the Goedel sentence.


The part that I am quoting has to do with the construction of the Tarksi sentence:
(3) x ? Pr ? x ? Tr

which is an enormously simplified analog to the Gödel sentence.
It is very obvious that it is not a Gödel sentence when it is
compared to the actual Gödel sentence from Gödel's actual paper.

(n)[Sb (a <sup>v</sup><sub>Z(n)</sub>) ? Flg(c)] & [Neg(v Gen a)] ? Flg(c)
I had to use HTML to indicate the subscripts and superscripts.

When I show that the Tarski sentence is semantically incoherent
then the Tarski proof that follows for the Tarski sentence is also
shown to be semantically incoherent.

> This substitution for x is not legitimate; x has to be the Goedel sentence.


x is not a Gödel sentence. x is analogous to the free variable v from
the above quoted Gödel sentence.

>>> The first step in the real proof is the diagonal lemma.

>> The diagonal lemma proves that the Tarski sentence cannot be satisfied.

> There is no such thing as "the Tarski sentence".


This is the Tarski sentence: (3) x ? Pr ? x ? Tr
[..]
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