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

On Mon, 2025-05-12 at 17:32 +0100, Ben Bacarisse wrote:
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Of course, there are no pathological inputs like this because H does not
exist.  

That is the key point.
The 'pathological' input D exists only when the assumed halting decider H exists.
D exists AFTER H does, so putting them together to discuss (or to form logical
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/download
...
Peano's axioms, which are the basis of mathematical logic, have a serious flaw:
Without a termination condition, it is impossible to explain "∞∉ℕ" (if the
definition of numbers relies on Peano's axioms).
This has led to many theories related to infinity, including density and other
laws and logical inferences, to have the same blind spots and mistakes (Personal
opinion: The reasoning/axiomatic system may need to be changed to a procedual
description)...

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

In short, my opinion is that TM/algorithm (C/C++/Assembly could and should be 
the language for building theorys of math/CS/..), is the foundation of science
better than contemporary, what-so-called 'strict, consistent/complete' axiomized
system.
Also, this idea can prove Church–Turing thesis: TM cannot be proved to be
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.