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*Every implementation of Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ gets the wrong answer*You agree. And since every implementation of Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ gets the same answer as H ⟨Ĥ⟩ ⟨Ĥ⟩ (it is stipulated) then obviously H ⟨Ĥ⟩ ⟨Ĥ⟩ also gets the wrong answer.
so what is wrong with it?It is also true that every instance of that question has a right answer, it just isn't the one that H gives.Every decision problem that includes undecidable instances only has
>
these instances because there is something wrong with the decision
problem specification.
The proof of the halting problem assumes a universal haltIf you tell me a consistent universal halt test I will tell you a consistent program that the test cannot handle. It will definitely be a program. There will be no valid rebuttal that it isn't a program at all.
test exists and then provides S as an example of a program
that the test cannot handle. But S is not a program at all.
It is not even a conceptual object, and this is due to
inconsistencies in the specification of the halting function.
(Stoddart: 2017)
*This is my unique contribution to the field of the Halting Problem*We do not ask it that. We ask it whether Ĥ halts on input ⟨Ĥ⟩. This is an objective specification, not subjective.
*This is my unique contribution to the field of the Halting Problem*
*This is my unique contribution to the field of the Halting Problem*
When we ask H ⟨Ĥ⟩ ⟨Ĥ⟩:
Does your input halt on its input?
meaning: Would Ĥ ⟨Ĥ⟩ halt on its input, then H gets the wrong answer.We can't ask that question because it is subjective. And that is a different question.
When we ask the exact same question meaning:
Will you halt if you never abort your simulation?
Then every H always gets the right answer.If you ask me to recite Einstein's equations, I will answer wrong. If you ask me 1+1, I will answer right. That I answer correctly to one question has no relevance to that I answer incorrectly to another question.
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