Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

On 5/7/2025 2:31 PM, olcott wrote:

On 5/7/2025 1:14 PM, Richard Heathfield wrote:

On 07/05/2025 18:55, olcott wrote:

When THERE IS NO CONTRADICTION then proof by contradiction fails.
How do you not get that?

>
I do. You must be talking about the Olcott Problem again, because the contradiction is inherent in the Halting Problem.
>

 Not when its terrible mistake is corrected.
 

It starts with the assumption that a universal halt decider can be written, and then shows that such a decider can be used to devise a program that the 'universal' decider can't decide --- a contradiction.
>
But you already know all this.
>

 I already know that the contradictory part of the
counter-example input has always been unreachable code.
 Thus PROOF BY CONTRADICTION FAILS because there never
was any actual contradiction.

False.  The contradiction is that HHH is found to not map the below function after it is assumed that the below function is computable and that HHH computes it:
Given any algorithm (i.e. a fixed immutable sequence of instructions) X described as <X> with input Y:
A solution to the halting problem is an algorithm H that computes the following mapping:
(<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
(<X>,Y) maps to 0 if and only if X(Y) does not halt when executed directly