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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Mon, 24 Mar 2025 22:29:06 -0500
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On 3/24/2025 10:12 PM, dbush wrote:
> On 3/24/2025 10:07 PM, olcott wrote:
>> On 3/24/2025 8:46 PM, André G. Isaak wrote:
>>> On 2025-03-24 19:33, olcott wrote:
>>>> On 3/24/2025 7:00 PM, André G. Isaak wrote:
>>>
>>>>> In the post you were responding to I pointed out that computable 
>>>>> functions are mathematical objects.
>>>>
>>>> https://en.wikipedia.org/wiki/Computable_function
>>>>
>>>> Computable functions implemented using models of computation
>>>> would seem to be more concrete than pure math functions.
>>>
>>> Those are called computations or algorithms, not computable functions.
>>>
>>
>> https://en.wikipedia.org/wiki/Pure_function
>> Is another way to look at computable functions implemented
>> by some concrete model of computation.
>>
> 
> And not all mathematical functions are computable, such as the halting 
> function.
> 
>>> The halting problems asks whether there *is* an algorithm which can 
>>> compute the halting function, but the halting function itself is a 
>>> purely mathematical object which exists prior to, and independent of, 
>>> any such algorithm (if one existed).
>>>
>>
>> None-the-less it only has specific elements of its domain
>> as its entire basis. For Turing machines this always means
>> a finite string that (for example) encodes a specific
>> sequence of moves.
> 
> False.  *All* turing machine are the domain of the halting function, and 
> the existence of UTMs show that all turning machines can be described by 
> a finite string.
> 

You just aren't paying enough attention. Turing machines
are never in the domain of any computable function.
<snip>

>>
>> The math halting function is free to "abstract away" key
>> details that change everything. That is why I have never
>> been talking about the pure math and have always been
>> referring to its implementation in a model of computation.
>>
> 
> There are no details abstracted away.  The halting function is simply 
> uncomputable.
> 

When the measure of the behavior specified by a
finite string input DD is when correctly emulated
by HHH then DD can't reach its own final halt state
then not-halting is decidable.

>> A halt decider cannot exist 
> 
> So again, you explicitly agree with the Linz proof and all other proofs 
> of the halting function.
> 
>> because the halting problem is defined incorrectly 
> 
> There's nothing incorrect about wanting something that would solve the 
> Goldbach conjecture and make unknowable truths knowable.  Your alternate 
> definition won't provide that.
> 
> There's no requirement that a function be computable.


-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer