> We programmers write halting programs with great regularity.
Making any program halting program is trivial: add executed instructions counter, halt program at some value of the counter. Proving that an arbitrary program halts is an entirely different task.
> So, the fact that we cannot solve some problems does not imply we are not halting oracles.
If it's allowed to not solve some problems, then I can write such an oracle:
Run a program for a million steps. If program has halted, output "Halts", otherwise output "Don't know".
It can't solve some problems, but by your logic it doesn't imply it's not a halting oracle. You are missing something.
Making any program halting program is trivial: add executed instructions counter, halt program at some value of the counter. Proving that an arbitrary program halts is an entirely different task.
> So, the fact that we cannot solve some problems does not imply we are not halting oracles.
If it's allowed to not solve some problems, then I can write such an oracle:
Run a program for a million steps. If program has halted, output "Halts", otherwise output "Don't know".
It can't solve some problems, but by your logic it doesn't imply it's not a halting oracle. You are missing something.