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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Halting Problem: How my refutation differs to Peter Olcott's
Date: Sun, 11 May 2025 04:23:00 +0100
Organization: Fix this later
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On 11/05/2025 03:51, Mr Flibble wrote:
> On Sat, 10 May 2025 21:49:41 -0400, Richard Damon wrote:
> 
>> On 5/10/25 9:18 PM, Mr Flibble wrote:
>>> On Sat, 10 May 2025 21:07:34 -0400, Richard Damon wrote:
>>>
>>>> On 5/10/25 9:00 PM, olcott wrote:
>>>>> On 5/10/2025 6:56 PM, Mr Flibble wrote:
>>>>>> On Sat, 10 May 2025 18:40:53 -0400, Richard Damon wrote:
>>>>>>
>>>>>>> On 5/10/25 4:38 PM, Mr Flibble wrote:
>>>>>>>> How my refutation differs to Peter's:
>>>>>>>>
>>>>>>>> * Peter refutes the halting problem based on pathological input
>>>>>>>> manifesting in a simulating halt decider as infinite recursion,
>>>>>>>> this being treated as non-halting.
>>>>>>>> * Flibble refutes the halting problem based on patholgical input
>>>>>>>> manifesting as decider/input self-referencial conflation,
>>>>>>>> resulting in the contradiction at the heart of the halting problem
>>>>>>>> being a category (type) error, i.e. ill-formed.
>>>>>>>>
>>>>>>>> These two refutations are related but not exactly the same.
>>>>>>>>
>>>>>>>> /Flibble
>>>>>>>
>>>>>>> And the problem is that you use incorrect categories.
>>>>>>>
>>>>>>> The decider needs to be of the category "Program".
>>>>>>>
>>>>>>> The input also needs to be of the category "Program", but provided
>>>>>>> via a representation. The act of representation lets us convert
>>>>>>> items of category Program to the category of Finite String which
>>>>>>> can be an input.
>>>>>>
>>>>>> Those two categories you have identified are different hence the
>>>>>> category error.
>>>>>>
>>>>>>
>>>>> That is correct. A running program and an input finite string ARE NOT
>>>>> THE SAME.
>>>>
>>>> But there is a direct relationship between the two.
>>>>
>>>>
>>>>
>>>>>>> The "Pathological Input" *IS* a Program, built by the simple rules
>>>>>>> of composition that are allowed in the system.
>>>>>>
>>>>>> Such composition is invalid.
>>>>>>
>>>>>>
>>>>> Richard is trying to get away with saying that a finite string THAT
>>>>> IS NOT A RUNNING PROGRAM <IS> A RUNNING PROGRAM
>>>>>
>>>>>
>>>> But they are related to each other,
>>>
>>> Even if there is some perceived relationship between the two different
>>> categories it doesn't mean there still isn't a category error.
>>
>> So, what is the error, since the input *IS* the finite string that was
>> built by the program representation operation, and thus *IS* what an
>> input needs to be.
>>
>>
>>> Why relationship doesn’t rescue the mistake:
>>>
>>> * Shared context ≠ shared type.
>>> – A pupil and a teacher are clearly related (one teaches, one learns),
>>> but the question “Who is taller, the lesson?” commits a category error
>>> because a lesson isn’t the kind of thing that has height, regardless of
>>> its pedagogical ties to people.
>>
>> Which doesn't apply here, and you are just indicationg you don't
>> understand what a representation is.
>>
>> The input is a finite string that represents a program.
> 
> A program and a finite string representing a program are different
> categories ergo we have a category error.

→ look

You are inside a halting problem, a category error or a proof.

There are some contradictions on the ground here.

There is a shiny brass Turing machine nearby.

There is a ball of finite string here.

There is a compiler error here.

→ take Turing machine

OK

→ Tie finite string to compiler error

OK

→ follow contradictions

You are in a maze of twisty little set theories, all different.

→ take category error

You are in a maze of twisty little set theories, all alike.

→ drink Entscheidungsproblem

I see no Entscheidungsproblem here.

→ stop or go to start?

You are inside a halting problem, a category error or a proof.

There are some contradictions on the ground here.

There is a ball of finite string here.

There is a compiler error here.

→

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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