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From: wij <wyniijj5@gmail.com>
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Subject: Re: =?UTF-8?Q?Flibble=E2=80=99s?= Leap: Why Behavioral Divergence
 Implies a Type Distinction in the Halting Problem
Date: Tue, 13 May 2025 06:57:43 +0800
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On Mon, 2025-05-12 at 17:32 +0100, Ben Bacarisse wrote:
[cut]
>=20
> Of course, there are no pathological inputs like this because H does not
> exist.=C2=A0=C2=A0

That is the key point.
The 'pathological' input D exists only when the assumed halting decider H e=
xists.
D exists AFTER H does, so putting them together to discuss (or to form logi=
cal
expression or even 'theorem', maybe) as though both exist at the same time =
is
very dubious.

Tradition logic is insufficient for math/logic, the foundation of science.
The implication is profound and goes beyond most people thought.

https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/d=
ownload
....
Peano's axioms, which are the basis of mathematical logic, have a serious f=
law:
Without a termination condition, it is impossible to explain "=E2=88=9E=E2=
=88=89=E2=84=95" (if the
definition of numbers relies on Peano's axioms).
This has led to many theories related to infinity, including density and ot=
her
laws and logical inferences, to have the same blind spots and mistakes (Per=
sonal
opinion: The reasoning/axiomatic system may need to be changed to a procedu=
al
description)...

So, in some POOH's rebuttal (just olcott's text, he don't understand what l=
ogic
'if' means) that uses logic (Tarski,Sisper,Russell,PL,...), that logic=20
themselves are invalid (I should save the long ellaberation).

In short, my opinion is that TM/algorithm (C/C++/Assembly could and should =
be=C2=A0
the language for building theorys of math/CS/..), is the foundation of scie=
nce
better than contemporary, what-so-called 'strict, consistent/complete' axio=
mized
system.
Also, this idea can prove Church=E2=80=93Turing thesis: TM cannot be proved=
 to be=20
exceeded simply because (current) axiomized system is less powerful than TM=
..

Back to POOH's pathological input theory:
The pathological input exists is because POOH exists.