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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 14:02:08 -0500
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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⟩

>> 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.

> 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.


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