int D(int (*x)())
{ int Halt_Status = H(x, x);
if (Halt_Status)
HERE: goto HERE;
return Halt_Status;
}

int main()
{ Output("Input_Halts = ", H(D,D));
}

When the job of a halt decider is to divide up a set of finite string
pairs into those that halt on their input and those that do not and one
of these inputs H is defined to do the opposite of whatever H returns we
have the exact same self-contradiction as the liar paradox.

When we ask whether this sentence is true or false:
"This sentence is not true" we find that it is not a truth bearer.
Any expression of language that is not in the category of truth bearer
has no truth values.

When we ask what halt status of 0 or 1 can H correctly return to D we
find that this question is also not a truth bearer.

"What time is it (yes or no)?"
shows an example of category error much more starkly. The answer is
required to come from the intersection of two disjoint sets.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 2/18/23 1:09 PM, olcott wrote:
> int D(int (*x)())
> {
>   int Halt_Status = H(x, x);
>   if (Halt_Status)
>     HERE: goto HERE;
>   return Halt_Status;
> }
>
> int main()
> {
>   Output("Input_Halts = ", H(D,D));
> }
>
> When the job of a halt decider is to divide up a set of finite string
> pairs into those that halt on their input and those that do not and one
> of these inputs H is defined to do the opposite of whatever H returns we
> have the exact same self-contradiction as the liar paradox.

Nope. Because the program DOES either halt or not, so there is a correct
answer, it just can't be given by THAT decider.

>
> When we ask whether this sentence is true or false:
> "This sentence is not true" we find that it is not a truth bearer.
> Any expression of language that is not in the category of truth bearer
> has no truth values.

Right, because the SENTENCE refers to itself.

>
> When we ask what halt status of 0 or 1 can H correctly return to D we
> find that this question is also not a truth bearer.

Except that isn't the question, but you seem to stupid to understand.

Remember, the machine claimed to be a Halt Decider must exist FIRST, and
must be willing to make the claim that it can process ANY input and
decide on it.

The "Pathological" program has definite behavior, it WILL either Halt or
run forever, and that behavior will depend on the answer that the halt
decider gives (doing the opposite).

The is nothing "invalid" about the way the "Pathological" program is
constructed, it is based on the same fundamental builing blocks as any
program is built on. All it does is show that the machine that claimed
it could handle ANY input was incorrect.

This "Pathological" program has well defined behavior, if H(P,P) will
return 1 then P(P) will Halt, otherwise it runs forever.

The "Pathological" structure is allowed by the basic rules of computations.

Your problem rests in thinking you can try to "tweek" your decider after
defining it but still let it be the same computation. This is not
allowed, once defined a machine/program is what it is and does what it
does. If you tweek it, you get a NEW and DIFFERENT program.

>
> "What time is it (yes or no)?"
> shows an example of category error much more starkly. The answer is
> required to come from the intersection of two disjoint sets.
>

Right, a question with a category error, which Halting doesn't have.