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From: olcott <polcott2@gmail.com>
Newsgroups: comp.theory,sci.logic
Subject: Re: ZFC solution to incorrect questions: reject them
Date: Tue, 12 Mar 2024 17:53:37 -0500
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On 3/12/2024 5:30 PM, Richard Damon wrote:
> On 3/12/24 2:34 PM, olcott wrote:
>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>> On 3/12/24 1:11 PM, olcott wrote:
>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>> On 3/12/24 12:02 PM, 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⟩
>>>>>
>>>>> No, because a given H will only go to one of the answers. THAT will 
>>>>> be wrong, and the other one right.
>>>>>
>>>>
>>>> ∀ H ∈ Turing_Machine_Deciders
>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>
>>>> Not exactly. A pair of otherwise identical machines that
>>>> (that are contained within the above specified set)
>>>> only differ by return value will both be wrong on the
>>>> same pathological input.
>>>
>>> You mean a pair of DIFFERENT machines. Any difference is different.
>>
>> Every decider/input pair (referenced in the above set) has a
>> corresponding decider/input pair that only differs by the return
>> value of its decider.
> 
> Nope.
> 
∀ H ∈ Turing_Machines_Returning_Boolean
∃ TMD ∈ Turing_Machine_Descriptions  |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)

Every H/TMD pair (referenced in the above set) has a
corresponding H/TMD pair that only differs by the return
value of its Boolean_TM.

That both of these H/TMD pairs get the wrong answer proves that
their question was incorrect because the opposite answer to the
same question is also proven to be incorrect.


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