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From: olcott <polcott2@gmail.com>
Newsgroups: comp.theory,sci.logic
Subject: =?UTF-8?Q?Re=3A_ZFC_solution_to_incorrect_questions=3A_reject_them_?=
 =?UTF-8?Q?--G=C3=B6del--?=
Date: Tue, 12 Mar 2024 15:35:27 -0500
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On 3/12/2024 3:31 PM, immibis wrote:
> On 12/03/24 20:02, olcott wrote:
>> On 3/12/2024 1:31 PM, immibis wrote:
>>> On 12/03/24 19:12, olcott wrote:
>>>> ∀ H ∈ Turing_Machine_Deciders
>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>
>>>> There is some input TMD to every H such that
>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>
>>> And it can be a different TMD to each H.
>>>
>>>> When we disallow decider/input pairs that are incorrect
>>>> questions where both YES and NO are the wrong answer
>>>
>>> Once we understand that either YES or NO is the right answer, the 
>>> whole rebuttal is tossed out as invalid and incorrect.
>>>
>>
>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does not halt
>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>
> 
> Once we understand that either YES or NO is the right answer, the whole 
> rebuttal is tossed out as invalid and incorrect.
> 
>>>> Does the barber that shaves everyone that does not shave
>>>> themselves shave himself? is rejected as an incorrect question.
>>>
>>> The barber does not exist. 
>>
>> Russell's paradox did not allow this answer within Naive set theory.
> 
> Naive set theory says that for every predicate P, the set {x | P(x)} 
> exists. This axiom was a mistake. This axiom is not in ZFC.
> 
> In Turing machines, for every non-empty finite set of alphabet symbols 
> Γ, every b∈Γ, every Σ⊆Γ, every non-empty finite set of states Q, every 
> q0∈Q, every F⊆Q, and every δ:(Q∖F)×Γ↛Q×Γ×{L,R}, ⟨Q,Γ,b,Σ,δ,q0,F⟩ is a 
> Turing machine. Do you think this is a mistake? Would you remove this 
> axiom from your version of Turing machines?
> 
> (Following the definition used on Wikipedia: 
> https://en.wikipedia.org/wiki/Turing_machine#Formal_definition)
> 
>>> The following is true statement:
>>>
>>> ∀ Barber ∈ People. ¬(∀ Person ∈ People. Shaves(Barber, Person) ⇔ 
>>> ¬Shaves(Person, Person))
>>>
>>> The following is a true statement:
>>>
>>> ¬∃ Barber ∈ People. (∀ Person ∈ People. Shaves(Barber, Person) ⇔ 
>>> ¬Shaves(Person, Person))
>>>
>>
>> That might be correct I did not check it over and over
>> again and again to make sure.
>>
>> The same reasoning seems to rebut Gödel Incompleteness:
>> ...We are therefore confronted with a proposition which
>> asserts its own unprovability. 15 ... (Gödel 1931:43-44)
>> ¬∃G ∈ F | G := ~(F ⊢ G)
>>
>> Any G in F that asserts its own unprovability in F is
>> asserting that there is no sequence of inference steps
>> in F that prove that they themselves do not exist in F.
> 
> The barber does not exist and the proposition does not exist.
> 

When we do this exact same thing that ZFC did for self-referential
sets then self-referential (thus pathological) inputs to halt deciders
cannot exist.

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