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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_o?=
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Date: Thu, 25 Jul 2024 10:51:24 -0500
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On 7/25/2024 3:55 AM, Mikko wrote:
> On 2024-07-23 14:53:21 +0000, olcott said:
> 
>> On 7/23/2024 3:07 AM, Mikko wrote:
>>> On 2024-07-22 14:40:41 +0000, olcott said:
>>>
>>>> On 7/22/2024 3:14 AM, Mikko wrote:
>>>>> On 2024-07-21 13:20:04 +0000, olcott said:
>>>>>
>>>>>> On 7/21/2024 4:27 AM, Mikko wrote:
>>>>>>> On 2024-07-20 13:22:31 +0000, olcott said:
>>>>>>>
>>>>>>>> On 7/20/2024 3:42 AM, Mikko wrote:
>>>>>>>>> On 2024-07-19 13:48:49 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Some undecidable expressions are only undecidable because
>>>>>>>>>> they are self contradictory. In other words they are undecidable
>>>>>>>>>> because there is something wrong with them.
>>>>>>>>>
>>>>>>>>> Being self-contradictory is a semantic property. Being 
>>>>>>>>> uncdecidable is
>>>>>>>>> independent of any semantics.
>>>>>>>>
>>>>>>>> Not it is not. When an expression is neither true nor false
>>>>>>>> that makes it neither provable nor refutable.
>>>>>>>
>>>>>>> There is no aithmetic sentence that is neither true or false. If 
>>>>>>> the sentnece
>>>>>>> contains both existentia and universal quantifiers it may be hard 
>>>>>>> to find out
>>>>>>> whether it is true or false but there is no sentence that is 
>>>>>>> neither.
>>>>>>>
>>>>>>>>  As Richard
>>>>>>>> Montague so aptly showed Semantics can be specified syntactically.
>>>>>>>>
>>>>>>>>> An arithmetic sentence is always about
>>>>>>>>> numbers, not about sentences.
>>>>>>>>
>>>>>>>> So when Gödel tried to show it could be about provability
>>>>>>>> he was wrong before he even started?
>>>>>>>
>>>>>>> Gödel did not try to show that an arithmetic sentence is about 
>>>>>>> provability.
>>>>>>> He constructed a sentence about numbers that is either true and 
>>>>>>> provable
>>>>>>> or false and unprovable in the theory that is an extension of 
>>>>>>> Peano arithmetics.
>>>>>>>
>>>>>>
>>>>>> You just directly contradicted yourself.
>>>>>
>>>>> I don't, and you cant show any contradiction.
>>>>>
>>>>
>>>> Gödel's proof had nothing what-so-ever to do with provability
>>>> except that he proved that g is unprovable in PA.
>>>
>>> He also proved that its negation is unprovable in PA. He also proved
>>> that every consistent extension of PA has a an sentence (different
>>> from g) such that both it and its negation are unprovable.
>>>
>>
>> L is the language of a formal mathematical system.
>> x is an expression of that language.
>>
>> When we understand that True(L,x) means that there is a finite
>> sequence of truth preserving operations in L from the semantic
>> meaning of x to x in L, then mathematical incompleteness is abolished.
> 
> No, it is not. From the meaning of "formal mathematical system" follows

I am overriding and superseding that. It is ridiculously
stupid to ignore the semantics of a formal expression
when evaluating its truth. Because truth *is* a semantic
property. Haskell Curry shows the correct way to evaluate
the semantic truth of an expression syntactically.
http://www.liarparadox.org/Haskell_Curry_45.pdf

Richard Montague's grammar of natural language semantics
does this same thing.

> that whether x is an expression of language L does not depend on semantics
> or L is not a language of a formal mathiematical system. In addition,
> the system is incomplete if there is a sentence that can be determined
> to be true from the meaning of x but cannot be proven in the system.
> 

This sentence is not true: "This sentence is not true" is only
true because the indirect reference of the outer sentence escapes
the self-contradiction. Tarski made this same stupid mistake:

Tarski's Liar Paradox from page 248
    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.
    https://liarparadox.org/Tarski_247_248.pdf

Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf

In my own Minimal Type Theory the self-reference would not
be so clumsy--- x := ~True(x)

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer