[Doctor Stochastic]

You're right about CH. It does seem to be in the same class as the parallel postulate. You can take it or leave it. I've never worked out the Goedel number for CH (or parallel) so I don't know how it fits in.

Basically Goedel maps all proofs into natural numbers then uses a variant of the Empedocles paradox. Goedel shows that the theorem: "This theorem cannot be proved," has a Goedel number the theorem is expressable in arithmetic; but it's proof isn't.

[longshadow]

I've never worked out the Goedel number for CH (or parallel) so I don't know how it fits in.

Correct me if I'm wrong here, but the Parallel Postulate isn't a Gödel statement, because geometry is exempt from Gödel's Incompleteness Theorem (because the axioms of Geometry don't incorporate the axioms for arithmetic on Natural numbers).

It is worthy of keeping in mind, it seems to me, that the essence of the Gödel's Incompleteness Theorem is that "there are some 'truths' about arithmetic of natural numbers we can't prove or disprove using only the axioms for arithmetic of Natural numbers..." The Parallel Postulate isn't a "truth" about arithmetic of Natural Numbers.

It also occurs to me that just because Gödel proved all formal systems that include the axioms of arithmetic of natural numbers will contain undecideable statements, it does not follow that ALL undecideable Mathematical statements are Gödelian in nature, the Euclidean Postulate being one counterexample.