> The independence of the parallel postulate doesn't make anyone > worry, because no one says you can carry out the proof of Fermat > in Euclidean geometry. The independence of the continuum > hypothesis says we really don't understand sets, and that's a worry.
It says we don't completely understand sets, but it also says most of the time, we don't need to. ZFC, which cannot settle the the continuum question, is actually stronger than we need most of the time. Most of the time, second order arithmetic would be fine. While in fact FLT was not proven in a system that weak, it's extremely likely that it could be, and we know that for a huge bunch of mathematics, it suffices.
On the other hand, sometimes painful issues do arise. The Whitehead problem, whether or not Ext^1(A, Z)=0 for A an abelian group entails that A is free, was a shocker to algebraists, you may recall. We would like to think what seem like basic algebraic questions of that nature are independent of set theory considerations--and usually, they are. But not this time, and in fact the whole question is bound up with the continuum hypothesis.
Cabal-type intutions, that the continuum hypothesis is false and Martin's axiom true, lead to one answer, whereas the minimalist hypothesis that all sets are constructible leads to another. You get the exact same division over Suslin lines; there are two very different flavors of set theory extensions which these reflect.