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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: A different perspective on undecidability --- incorrect question
Date: Sat, 26 Oct 2024 07:59:33 -0500
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On 10/26/2024 2:52 AM, Mikko wrote:
> On 2024-10-25 14:37:19 +0000, olcott said:
> 
>> On 10/25/2024 3:14 AM, Mikko wrote:
>>> On 2024-10-24 16:07:03 +0000, olcott said:
>>>
>>>> On 10/24/2024 9:06 AM, Mikko wrote:
>>>>> On 2024-10-22 15:04:37 +0000, olcott said:
>>>>>
>>>>>> On 10/22/2024 2:39 AM, Mikko wrote:
>>>>>>> On 2024-10-22 02:04:14 +0000, olcott said:
>>>>>>>
>>>>>>>> On 10/16/2024 11:37 AM, Mikko wrote:
>>>>>>>>> On 2024-10-16 14:27:09 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> The whole notion of undecidability is anchored in ignoring the 
>>>>>>>>>> fact that
>>>>>>>>>> some expressions of language are simply not truth bearers.
>>>>>>>>>
>>>>>>>>> A formal theory is undecidable if there is no Turing machine that
>>>>>>>>> determines whether a formula of that theory is a theorem of that
>>>>>>>>> theory or not. Whether an expression is a truth bearer is not
>>>>>>>>> relevant. Either there is a valid proof of that formula or there
>>>>>>>>> is not. No third possibility.
>>>>>>>>>
>>>>>>>>
>>>>>>>> After being continually interrupted by emergencies
>>>>>>>> interrupting other emergencies...
>>>>>>>>
>>>>>>>> If the answer to the question: Is X a formula of theory Y
>>>>>>>> cannot be determined to be yes or no then the question
>>>>>>>> itself is somehow incorrect.
>>>>>>>
>>>>>>> There are several possibilities.
>>>>>>>
>>>>>>> A theory may be intentionally incomplete. For example, group theory
>>>>>>> leaves several important question unanswered. There are infinitely
>>>>>>> may different groups and group axioms must be true in every group.
>>>>>>>
>>>>>>> Another possibility is that a theory is poorly constructed: the
>>>>>>> author just failed to include an important postulate.
>>>>>>>
>>>>>>> Then there is the possibility that the purpose of the theory is
>>>>>>> incompatible with decidability, for example arithmetic.
>>>>>>>
>>>>>>>> An incorrect question is an expression of language that
>>>>>>>> is not a truth bearer translated into question form.
>>>>>>>>
>>>>>>>> When "X a formula of theory Y" is neither true nor false
>>>>>>>> then "X a formula of theory Y" is not a truth bearer.
>>>>>>>
>>>>>>> Whether AB = BA is not answered by group theory but is alwasy
>>>>>>> true or false about specific A and B and universally true in
>>>>>>> some groups but not all.
>>>>>>
>>>>>> See my most recent reply to Richard it sums up
>>>>>> my position most succinctly.
>>>>>
>>>>> We already know that your position is uninteresting.
>>>>>
>>>>
>>>> Don't want to bother to look at it (AKA uninteresting) is not at
>>>> all the same thing as the corrected foundation to computability
>>>> does not eliminate undecidability.
>>>
>>> No, but we already know that you don't offer anything interesting
>>> about foundations to computability or undecidabilty.
>>
>> In the same way that ZFC eliminated RP True_Olcott(L,x)
>> eliminates undecidability. Not bothering to pay attention
>> is less than no rebuttal what-so-ever.
> 
> No, not in the same way. 

Pathological self reference causes an issue in both cases.
This issue is resolved by disallowing it in both cases.

When we disallow the Liar Paradox then Tarski cannot derive
the first state of his proof and his proof fails.

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

adapted to become this
x ∉ Pr if and only if p  // line 1 of the proof

Here is the Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p       // assumption (see above)
(2) x ∈ True if and only if p              // Tarski's convention T
(3) x ∉ Provable if and only if x ∈ True. // (1) and (2) combined
(4) either x ∉ True or x̄ ∉ True;      // axiom: ~True(x) ∨ ~True(~x)
(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

> ZFC is a useful set theory for many purposes.
> You don't offer any useful theory for any purpose.
> 

If we had a True(L, x) that worked consistently and L
is formalized natural language then we could refute
all of the dangerous lies made for political gain in
real time before they gained any traction.

Because we don't have this it looks like there is a
good chance we will be seeing the rise of the Fourth
Reich in a few days.

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