Jiri Lebl wrote: > 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.
And this changes your practice of mathematics, exactly how?
> 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.
On the contrary, it knows nothing else but the axioms you as subject put at-play for it as object.
> 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.
But, that's exactly what agreement means.
> 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.
What I said was that believing in set theory means you agree with the axioms of set theory. Where is your notion of belief coming from?