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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 22:14:15 -0500
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On 3/12/2024 9:51 PM, Richard Damon wrote:
> 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.
> 

*Not in my actual example where F has its own provability operator*

>>
>> 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.
> 
No one can prove that they themselves do not exist.
Thus G cannot possibly derive a sequence of inference
steps that prove that they themselves do not exist.

> 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.
> 
I am not referring to that one.

> 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.
> 
No formal system can possibly have any sequence of inference
step that prove that they themselves do not exist because
this is self-contradictory, not because the system is incomplete.
In my G this is obvious.

Whether or not Gödel's G is isomorphic to mine I have proved
this this is false by providing the counter-example of my G.

....14 Every epistemological antinomy can likewise be used for a similar 
undecidability proof...(Gödel 1931:43-44)

> 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.
> 

Or it could be the epistemological antinomy of trying to find a
sequence of inference steps in F that prove that they themselves
do not exist.
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