Re: The halting problem as defined is a category error

On 7/23/2025 2:34 AM, Mikko wrote:

On 2025-07-22 13:56:36 +0000, olcott said:
 

On 7/22/2025 5:51 AM, Mikko wrote:

On 2025-07-21 14:07:27 +0000, olcott said:
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On 7/21/2025 4:38 AM, Mikko wrote:

On 2025-07-20 15:04:34 +0000, olcott said:
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On 7/20/2025 3:46 AM, Mikko wrote:

On 2025-07-19 14:59:41 +0000, olcott said:
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On 7/19/2025 4:02 AM, Mikko wrote:

On 2025-07-18 22:11:50 +0000, Mr Flibble said:
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On Thu, 17 Jul 2025 13:01:31 -0500, olcott wrote:
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Claude.ai agrees that the halting problem as defined is a category
error.
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https://claude.ai/share/0b784d2a-447e-441f-b3f0-a204fa17135a
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This can only be directly seen within my notion of a simulating halt
decider. I used the Linz proof as my basis.
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Sorrowfully Peter Linz passed away 2 days less than one year ago on my
Mom's birthday July 19, 2024.

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I was the first to state that the halting problem as defined is a category
error and I stated it in this forum.

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Indeed you stated that but failed to identify the actual error. You
did not say which word in the problem statement is wrong or what is
the wrong category or what would be the right one.

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I conclusively proved the actual category error yet
people that are only interested in rebuttal want no
part of any proof that I am correct.

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Is it the same error as Flibble found?

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Flibble's category error is stated abstractly.
My version is stated concretely.

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Could you post a pointer to your version?

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The category error is a type mismatch error where
a Turing Machine decider is required to report on
the behavior of a directly executed machine yet
cannot take a directly executed machine as an input.

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That is not a category error. A category error is a word or phrase
of some category in a context that requires a word or phrase of a
different category.

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The category error is the mistake of assuming that
a directly executing Turing machine is in the category
of input to a Turing machine halt decider.

 That error is not present in the halting problem. It is also not
present in https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
which is the prototype of proofs that you falsely claim to have
refuted.
 

*I am either going to go by the Linz proof*
*From the bottom of page 319 has been adapted to this*
https://www.liarparadox.org/Peter_Linz_HP_317-320.pdf
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.∞,
    if Ĥ applied to ⟨Ĥ⟩ halts, and
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
    if Ĥ applied to ⟨Ĥ⟩ does not halt.
*or my own code*
int DD()
{
   int Halt_Status = HHH(DD);
   if (Halt_Status)
     HERE: goto HERE;
   return Halt_Status;
}

The category error is typically stated indirectly by
requiring a Turing machine based halt decider to report
on the behavior of a directly executed Turing machine.

 An indirect statement of a category error is not a category error.
Category error is a word that does not fit in its context.
 

It can be easily corrected by changing the requirement
to report on the behavior that its finite string input
specifies.

 Then the word "speifies" needs a definition. The clearest formulation
is to ask about an Universal Turing machine. Then the input to the
decider is the same as the input tu the Universal Turing machine.
 

When an input calls its own decider in recursive emulation
this must be modeled. The best way to determine the behavior
that the input specifies is for DD to be emulated by HHH
according to the semantics of the x86 language. That does
cause recursive emulation.

*I have conclusively proven that these behaviors diverge*

 No, you have not. A Turing machine has only one behaviour and the
halting problem requires that the input to the halting decider
describes that behaviour. If you inteprete the input differently
then either the input or the interpretaion is wrong.
 

That is counter-factual yet you can not understand
how it is counter-factual until after you first study
the 3.5 page Linz proof.
https://www.liarparadox.org/Peter_Linz_HP_317-320.pdf

That people cannot understand this proof does not mean that
it is not a proof.

 A sufficeint understanding of your "proofs" is that they are
not what the word "proof" means.
 

I am going by the deductive model where verified
true premises with correct reasoning derive true
conclusions. Unlike most proofs mine are semantic
not merely syntactic.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer