[BruceZ]

I was really only concerned with the distinction of "P is neither true nor false" and "P cannot be proven to be true nor false"

"Proveability is a weaker notion than truth".

your claim after stating Russel's Paradox seemed to imply the paradox was the proof of the inconsistency of all axiomatic systems.

It led to a proof of the inconsistency of the axioms of mathematics (naive set theory) that existed at the time the paradox was discovered. It was a proof that any consistent axiomatic system (of complexity sufficient to support basic arithmetic) would necessarily be incomplete. So it was a proof of the inconsistency of complete axiomatic systems.

Being precise in these cases would not have led to a significantly longer (or less comprehensible) post.

The other issue was your statement that we cannot know that the axioms of mathematics are consistent. We can prove that the axioms of textbook mathematics (axiomatic set theory) are consistent, but only by using other axioms which create a larger system which then cannot prove itself consistent. Also, certain individual branches of mathematics can prove themselves to be consistent and complete. I wasn't sure if you were aware of these results. There also seemed to be a sudden interest in Gödel in general.

One thing I said that is probably not true is that the irrationality of the sqrt(2) is proveable within the consistent and complete theory of the real field. I doubt there can be any notion of irrationality in this theory since the integers cannot be constructed from this theory.