Questions about consistency
I was just learning about Baire Category from Royden's book today, and it got me thinking about consistency. Now here's my question:
1. Suppose Bob (some mathematician) constructs a counter-example to a theorem in real analysis which has already been proven rigorously from the axioms of the real numbers and ZFC. Then wouldn't it follow that ZFC set theory is inconsistent?
2. Is the situation in (1) even possible? That is, could such an inconsistency be found theoretically? Could the death knell counter-example to modern mathematics be lurking out there somewhere for somebody to find?
The way I understand it, no system can be proven consistent without embedding it in some larger system (I believe I was told that anyway, and it makes sense). If you can't prove it consistent, then you never know for *sure* that somebody won't someday find such an inconsistency, right? That leads me to
3. Do you, the mathematical community, believe that such a counter-example might someday be found?