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From: olcott <polcott2@gmail.com>
Newsgroups: comp.theory,sci.logic
Subject: Re: ZFC solution to incorrect questions: reject them (NFFC)
Date: Tue, 12 Mar 2024 16:42:10 -0500
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On 3/12/2024 4:29 PM, Richard Damon wrote:
> On 3/12/24 1:35 PM, olcott wrote:
>> 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.
>>
> 
> And your computation system isn't Turing Complete, by definition.
> 

I don't think that this has anything to do with Turing completeness.
It is more a matter of reestablishing the notion of computation on
a new foundation the same way that ZFC did for Naive set theory.

ZFC removed logically impossible decision problem instances. My new
foundation for computation (NFFC) only removes logically impossible
decision problem instances. Turing machines remain the same.

> You are saying there are some computations that Turing Machines can do 
> (like H^) that can not exist in your system.
> 
> That option has been rejected as viable for Computation Theory

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