Re: Flibble’s Leap: Why Behavioral Divergence Implies a Type Distinction in the Halting Problem

Richard Heathfield <rjh@cpax.org.uk> writes:

On 12/05/2025 17:32, Ben Bacarisse wrote:

...

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

This was a very common problem with students.  They so buy into the
assumption "let H be a TM that decides halting" that they think that H'
and H^ (to use Linz's notation) also exist and so there really is a
concrete string of symbols (the encoding of H^ which I write as [H^])
such that H^([H^]) halts if it doesn't and doesn't halt if it does.
Of course, there are no pathological inputs like this because H does not
exist.  For every TM, M, the machines M' and M^ do indeed exist, [M^]
really is some finite string of symbols and M^([M^]) either halts or it
does not halt.  But because none of the infinite variety of Ms
implements the halting function, there is no paradox or pathological
input anywhere to be seen.

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>
Here, Ben cuts straight to the heart of the matter.

You are too kind.

...

Aside...  Forgive me for repeatedly replying to you.

>
Nothing to forgive.
>

It's because I
know you from comp.lang.c and because you are, I think, new to this
thread which has been going on for over 20 years.

>
I am, yes. 20 years, though? Really? That's one hell of a filibuster, albeit
one that's several decades too late.

The first record I have in my notes is from 2004:

From: "Peter Olcott" <olc...@worldnet.att.net>
Subject: Re: Attempt to Refute the Halting Problem's Refutation
Message-ID: <3qbTc.205522$OB3.200659@bgtnsc05-news.ops.worldnet.att.net>
Date: Fri, 13 Aug 2004 22:38:55 GMT

Statement of what I have achieved
I have correctly refuted each and every mechanism by which the
above statement has been proven to be true. I have not shown that
solving the Halting Problem is possible, merely refuted every proof
that it is impossible.

...which suggests that he may be yanking a 20-year-old chain just because he
likes to hear it rattle.

I would suggest that he is the kind of narcissist that gets his
validation from the status of the people he insults.  I think there is a
name for it, but I forget.  Usenet is their natural home.  Einstein was
wrong, Cantor was wrong, Turing was wrong...

--
Ben.