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

On 2024-03-15 14:28:49 +0000, olcott said:

On 3/15/2024 6:07 AM, Mikko wrote:

On 2024-03-13 14:25:10 +0000, olcott said:
 

On 3/13/2024 5:03 AM, Mikko wrote:

On 2024-03-12 20:38:34 +0000, olcott said:
 

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 Gödel's self-referential expressions that assert their
own unprovability in F also cease to exist.

 Although Russel's set cannot be costructed in in ZFC Gödel's set can,
thus proving that ZFC is incomplete and ZFC augmented with additional
axioms is either incomplete or inconsistent.
 

 That is not how it works at all. Russell's paradox pointed out
incoherence in the notion of a set. ZFC fixed that.
 The inability to show the a self-contradictory sentence is true or false
is merely the inability to do the logically impossible and places no
actual limit on anyone or anything.

 If a theory is complete there is a simple computable method to find out
whether a particular sentence is a theorem or not. That method does not
work with incomplete theories, and in many cases, including ZF and ZFC,
no method works.
 

 To say that anything or anyone is in anyway limited or incomplete
because they lack the ability to do the logically impossible is
incorrect.
 Human knowledge is not incomplete on the basis of the lack of the ability to prove the Liar Paradox is true or false.
"This sentence is not true." is not true, yet neither true nor false.
 The Liar Paradox is not truth bearer thus has no truth value.
Tarski concluded that True(L,x) cannot be defined because it
gets stumped on the Liar Paradox.

Every sentence in the language of ZFC either is or is not provable.
Anything else is a logical impossibility.
There is no complete method to determine whether a sentence in the
language of ZFC is provable. Existence of such method is a logical
impossibility.
--
Mikko