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From: joes <noreply@example.org>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 08:32:25 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Mon, 24 Mar 2025 22:29:06 -0500 schrieb olcott:
> 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.
>>>>> 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>
Fine, their descriptions are, and their behaviour is computable - 
by running them.

>>> 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.
What details?

>> 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.
Circular reasoning. You'll have to prove HHH simulates correctly first.

>>> 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.
-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.