Karl Malbrain wrote: > By the definition I gave above, it is. When you work within the ZFC > system you have to agree to the axioms of ZFC. What is the difference > between "pragmatic acceptance" and belief?
There is a world of difference. If I truly believe something to be true then I do not admit the possibility of it being false. I do allow for the possibility that someone will find an inconsistency in ZFC. Thus I don't believe in ZFC. I find it improbable, but possible.
Further, a computer can construct a proof using ZFC. But the computer does not even know the "meaning" of the statements, it most certainly does not "believe" in the axioms. In fact I could in fact think that a given axiom system A is in fact bad and inconsistent, and still construct proofs A => B. For example my acceptance of axiom of choice is purely from the standpoint of "it makes many proofs easy, it proves wonderful results and it doesn't seem to mess things up too much" I don't particularly believe that it is true or not. I am happier with a proof not using axiom of choice and I'm happy with proofs assuming the negation of axiom of choice. Why do you want to force me to "believe" in something I clearly don't.
I'm starting to believe, however, that you are, in fact, a crank.