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From: Richard Damon <richard@damon-family.org>
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 19:51:26 -0700
Organization: i2pn2 (i2pn.org)
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On 3/12/24 4:14 PM, olcott wrote:
> 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)
>>>>>>>>>
>>>>>>>>> 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.
>>>>>
>>>>
>>>> And you end up with a very weak logic system that can't even have 
>>>> the full properties of the Natuarl Numbers.
>>>
>>> Natural numbers never really did have the property of provability.
>>> This was something artificially contrived that never really belonged
>>> to them.
>>>
>>
>> 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)

Right, seen in the meta-Theory from F.

> 
> based on immblis Russell's Paradox reply
> ¬∃G ∈ F | G ↔ ~(F ⊢ G)  // is simply false

Then a proof must exist in F that G is True, So G can't be false.

Note, The statement G is NOT a statement about itself being provable, 
that is only a semantic revealed in the RIGHT meta-theory.

> 
> 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.
> 

No FINITE sequence of inference steps.

Note, the key is that G doesn't assert that, G is a statement about 
math, that only when interpreted in a meta-theory sees that.

G is actually a statment about the existance of a number that matches a 
complected and carefully constructed relationship, that is fully 
computable. The Existance or non-existance of such a number is a pure 
binary thing, either it WILL or it WON'T.

The complicated relationship deals with a way to encode as a number ANY 
analyitic statement in F (since all statements are just strings, and 
strings can be encoded into a number), and a calculation to see if that 
statement actually is a proof starting with the enumerated truth makers 
of F, through the valid and allowed logical operatons to the statement 
of G. A number that satisfies this relation, encodes a proof of G, and 
any such proof, can always be encoded in to a number.

G is the statement that no such number exist. So, if G is false, then 
such a number exists, and then in the Meta-Theory, we can decode that 
number into a proof in F that G must be true.

If this is the case, then F must be inconsistant, and the proof starts 
with the presumption that we are dealing with a consistant logic system.

But, if G is true, then there can be no such number, and thus from our 
knowledge in the mete-theory, we know that no proof CAN exist for that fact.

Thus, the only possiblity for this statement, is to be True, but unprovable.

The big part of the theory is showing that such a relationship can be 
made, and that shows that proof checkers are a computable function.

Given a system, and an enumeration of its basic truths, we can check if 
a proof present actually proves the result it claims to.