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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: The philosophy of logic reformulates existing ideas on a new
 basis --- infallibly correct
Date: Mon, 11 Nov 2024 08:35:43 -0600
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On 11/11/2024 4:26 AM, Mikko wrote:
> On 2024-11-11 03:08:36 +0000, olcott said:
> 
>> On 11/10/2024 3:52 AM, Mikko wrote:
>>> On 2024-11-09 18:05:38 +0000, olcott said:
>>>
>>>> On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
>>>>> olcott <polcott333@gmail.com> wrote:
>>>>>> On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
>>>>>>> olcott <polcott333@gmail.com> wrote:
>>>>>>>> On 11/9/2024 5:01 AM, joes wrote:
>>>>>>>>>>>>>>>>> On 11/8/24 12:25 PM, olcott wrote:
>>>>>
>>>>>>>>>>>>>>>>>> That formal systems that only apply truth preserving 
>>>>>>>>>>>>>>>>>> operations
>>>>>>>>>>>>>>>>>> to expressions of their formal language that have been
>>>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable 
>>>>>>>>>>>>>>>>>> is proven
>>>>>>>>>>>>>>>>>> to be over-your-head on the basis that you have no actual
>>>>>>>>>>>>>>>>>> reasoning as a rebuttal.
>>>>>>>>> Gödel showed otherwise.
>>>>>
>>>>>>>> That is counter-factual within my precise specification.
>>>>>
>>>>>>> That's untrue - you don't have a precise specification.  And even 
>>>>>>> if you
>>>>>>> did, Gödel's theorem would still hold.
>>>>>
>>>>>>>> When truth is only derived by starting with
>>>>>>>> truth and applying truth preserving operations
>>>>>>>> then unprovable in PA becomes untrue in PA.
>>>>>
>>>>>>> No.  Unprovable will remain.
>>>>>
>>>>>> *Like I said you don't pay f-cking attention*
>>>>>
>>>>> Stop swearing.  I don't pay much attention to your provably false
>>>>> utterances, no.  Life is too short.
>>>>>
>>>>
>>>> That you denigrate what I say without paying attention to what
>>>> I say <is> the definition of reckless disregard for the truth
>>>> that loses defamation cases.
>>>>
>>>>> Hint: Gödel's theorem applies in any sufficiently powerful logical
>>>>> system, and the bar for "sufficiently powerful" is not high.
>>>>>
>>>>
>>>> Unless it is stipulated at the foundation of the notion of
>>>> formal systems that ~Provable(PA, g) simply means ~True(PA, g).
>>>>
>>>>>> Unprovable(L,x) means Untrue(L,x)
>>>>>> Unprovable(L,~x) means Unfalse(L,x)
>>>>>> ~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
>>>>>
>>>>> If you're going to change the standard meaning of standard words, 
>>>>> you'll
>>>>> find communicating with other people somewhat strained and difficult.
>>>>>
>>>>
>>>> ZFC did the same thing and that was the ONLY way
>>>> that Russell's Paradox was resolved.
>>>>
>>>> When ~Provable(PA,g) means ~True(PA,g) then
>>>> incompleteness cannot exist.
>>>
>>> But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA
>>> and "~Provable(PA,g)" means that there is not. These meanings are don't
>>> involve your "True" in any way. You may define "True" as a synonym to
>>> "Provable" but formal synonyms are not useful.
>>
>> We can ALWAYS prove that any expression of language is true or
>> not on the basis of other expressions of language when we have a
>> coherent definition of True(L,x).
> 
> Not relevant. 

It <is> relevant in that it does refute the Tarski
Undefinability theorem that <is> isomorphic to incompleteness.

> The meaning of "Provable(PA,g)" does not depend on
> the definition of "True(L,x)". "Provable(PA,g)" is false because
> there is no proof of g in PA. For the same reason "Provable(PA,~g)"
> is false.
> 

There is no proof of Tarski's x in his Theory only
because x is incoherent in his theory.
https://liarparadox.org/Tarski_275_276.pdf

    Let {T} be such a theory. Then the elementary
    statements which belong to {T} we shall call the
    elementary theorems of {T}; we also say that
    these elementary statements are true for {T}.
    Thus, given {T}, an elementary theorem is an
    elementary statement which is true.
    https://www.liarparadox.org/Haskell_Curry_45.pdf

Haskell Curry is referring to a set of expressions that are
stipulated to be true in T.

We define True(L, x) to mean x is a necessary consequence of
the Haskell Curry elementary theorems of L.
(Haskell_Curry_Elementary_Theorems(L) □ x) ≡ True(L, x)

x = "What time is it?"
True(English, x) == false
True(English, ~x) == false
∴ Not_a_Truth_Bearer(English, x)

Under math rules we would declare that English is incomplete
because neither x nor ~x is provable in English.

> There are actually infinitely many sentences of PA that could be used
> instead of g to show incompleteness but one is enoubh.
> 
>> That Gödel relies on True(meta-math, g) to mean True(PA, g)
>> is a stupid mistake that enables Incomplete(PA) to exist.
> 
> Gödel proved Provable(meta-math, "~Provable(PA,g) ∧ ~Provable(PA,g)").
> 

That is the same thing as proving:
This sentence is not true: "This sentence is not true" is true.

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