[gizmo686]

> The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen.

As an example of this, the Greeks assumed that every number was rational (actually, that any two numbers could be expressed as an integer multiple of some common unit). When they did prove that this was false, they re-worked math without it and found that (almost?) all of their other theorems still worked even without the assumption that all numbers were rational.