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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: A different perspective on undecidability --- incorrect question
Date: Sat, 26 Oct 2024 11:48:49 -0400
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On 10/26/24 8:59 AM, olcott wrote:
> 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.

Nope, because is set theory, the "self-reference" was only directly 
available in the definition of a set. By disallowing the set itself to 
be in the set of things it can contain, ZF eleminated the problem.

For computations we have the problem that BY DEFINITION they need to be 
able to handle "ANY" finite input string.

And, since any computation can be expressed as a finite string, we can 
not exclude as the input, a string that represents the program (or 
contains which incudes the program as part of the input).

The problem gets compounded in that there aren't just a "few" inputs 
that could repreesent the program,

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

But he shows that you can, and thus your claim fails. All you are saying 
is that if we take "all" to not mean "all" we might be able to do 
something, but since all does mean all, that can't apply.

> 
> 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
> 

By which he shows that the language itself has been shown to support the 
representation of the Liar's Paradox, and thus it *IS* a valid input.



> 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.

But Tarski shows that a True((L, x), defined to work on ALL x that are 
expressable in L, can not be defined, as there exists some x that it can 
not have a consistent value for.

Your logic requires that ALL doesn't actually mean ALL, and thus your 
logic system is just not consistently defined.

> 
> 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.
> 

You just don't understand cause and effect it seems.

It is YOUR type of thinking that is fueling those dangers, so consider 
yourself part of the problem.