Re: Correcting the definition of the halting problem --- Computable functions

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