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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
Subject: =?UTF-8?Q?Re=3A_Working_out_the_details_of_the_steps_of_=C4=A4=2EH_?=
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Date: Fri, 8 Mar 2024 20:28:17 -0800
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On 3/8/24 8:05 PM, olcott wrote:
> On 3/8/2024 9:48 PM, Richard Damon wrote:
>> On 3/8/24 7:14 PM, olcott wrote:
>>> On 3/8/2024 9:03 PM, Richard Damon wrote:
>>>> On 3/8/24 6:34 PM, olcott wrote:
>>>>> On 3/8/2024 8:24 PM, Richard Damon wrote:
>>>>>> The Mapping describes the answer that we want for ALL possible 
>>>>>> inputs. It becomes the specification of the problem.
>>>>>>
>>>>>
>>>>> You still can't understand how computing the mapping
>>>>> from all inputs to final state Ĥ.Hqn or non final state
>>>>> Ĥ.Hqy on the basis of the indirect criteria also causes
>>>>> H ⟨Ĥ⟩ ⟨Ĥ⟩ <H> to correctly compute halting.
>>>>>
>>>>
>>>> Because your "indirect Criteria" map differs from the DEFINED DIRECT 
>>>> Criteria map.
>>>>
>>> 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
>>>
>>> The indirect criteria provides Ĥ.H the basis for
>>> which wrong answer it must return and provides
>>> H with the basis to return the correct halt status.
>>
>> And if it gives a WRONG answer, then H is just WRONG.
>>
>>>
>>> So far everyone in world the has no idea what
>>> basis Ĥ.H could use to determine its wrong answer.
>>> *They leave it wide open with a question mark*
>>
>> But it needs to get the RIGHT answer to be correct.
> 
> *Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ <Ĥ> must have some basis for its wrong answer*
> *Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ <Ĥ> must have some basis for its wrong answer*
> *Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ <Ĥ> must have some basis for its wrong answer*
> 
> 

But if it is the wrong answer, it is wrong.

Why don't you understand that?

H^.H (H^) (H^) <H^> will do EXACTLY what the algorith for H, applied to 
that input tells it to do.

That creates a behavior which determines what answer H gives, and the 
behavior that H must answer about.

Given the algroithm has been defined (or else we don't HAVE an H so we 
don't have an H^ to be talking about) the FACT of the behavior is fixed, 
it will do what it will do. We can then evaluate if it got the right 
answer by comparing the answer that it gave, with the answer the mapping 
we are trying to compute says it should have given to be correct.

Since, by the pathological nature of H^, the answer that it gives will 
NEVER match the answer that it should have given, this H will just 
always be wrong.

The question isn't wrong, as the mapping tells us what the answer should 
have been. It just turns out that the mapping can't be computed because 
for ANY machine that we might try to make a decider, there WILL be an 
input corresponding to this contrary nature, and thus we can conclude 
that the Halting Mapping is just uncomputable, meaning no Computable 
Algorithm (like a Turing Machine) exists that can ALWAYs compute the 
right answer.