Re: ZFC solution to incorrect questions: reject them --Gödel--

On 3/12/2024 6:00 PM, Richard Damon wrote:

On 3/12/24 2:44 PM, olcott wrote:

On 3/12/2024 4:31 PM, Richard Damon wrote:

On 3/12/24 1:38 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)
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There is some input TMD to every H such that
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)

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And it can be a different TMD to each H.
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When we disallow decider/input pairs that are incorrect
questions where both YES and NO are the wrong answer

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Once we understand that either YES or NO is the right answer, the whole rebuttal is tossed out as invalid and incorrect.
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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⟩
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Once we understand that either YES or NO is the right answer, the whole rebuttal is tossed out as invalid and incorrect.
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Does the barber that shaves everyone that does not shave
themselves shave himself? is rejected as an incorrect question.

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The barber does not exist.

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Russell's paradox did not allow this answer within Naive set theory.

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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.
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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?
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(Following the definition used on Wikipedia: https://en.wikipedia.org/wiki/Turing_machine#Formal_definition)
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The following is true statement:
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∀ Barber ∈ People. ¬(∀ Person ∈ People. Shaves(Barber, Person) ⇔ ¬Shaves(Person, Person))
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The following is a true statement:
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¬∃ Barber ∈ People. (∀ Person ∈ People. Shaves(Barber, Person) ⇔ ¬Shaves(Person, Person))
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That might be correct I did not check it over and over
again and again to make sure.
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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)
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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.

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The barber does not exist and the proposition does not exist.
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When we do this exact same thing that ZFC did for self-referential
sets then Gödel's self-referential expressions that assert their
own unprovability in F also cease to exist.
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And you end up with a very weak logic system that can't even have the full properties of the Natuarl Numbers.

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Natural numbers never really did have the property of provability.
This was something artificially contrived that never really belonged
to them.
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 No, Godel showed (or maybe used a previous proof) that you can use the Mathematics of Natural Numbers to test if a proof is valid.
 You just don't understand it. It really is very related to how Turing Machines work, which can be converted to a mathematical model.
 There is a field that looks at the comparability of Computations to Logic, so they are all really quite related.

This is refuted.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ...(Gödel 1931:43-44)
based on immblis Russell's Paradox reply
¬∃G ∈ F | G ↔ ~(F ⊢ G)  // is simply false
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