Re: Working out the details of the steps of Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ <Ĥ> ⊢* Ĥ.Hqn

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.
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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.
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Because your "indirect Criteria" map differs from the DEFINED DIRECT Criteria map.
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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
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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.

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And if it gives a WRONG answer, then H is just WRONG.
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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*

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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.