> If the Goldbach Conjecture is true as a consequence of an infinite number
> of independent mathematical facts, it would mean that there's a *specific*
> problem that *no* Turing machine can solve.
That doesn't sound right, as you can always add the GC or some equivalent
theorem to your axiom base. I don't see how a fact like the GC can be
true only as a consequence of an infinite number of facts. It seems for
that to be true that your fact would have to carry an infinite amount of
information, like the value of Chaitin's Omega.
I still think it is useful to think of Euclidean geometry minus the
parallel postulate to gain insight into these issues. No TM is ever
going to prove the PP starting with this trucated axiom base. But I
don't think we would say that the PP's truth is the consequence of an
infinite number of independent geometrical facts; at least, no more so
than any other axiom.
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