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?

1. Yes, in that case ZFC would be inconsistent.

2. Yes, it is theoretically possible. As you yourself mention, if ZFC happens to be consistent, then it cannot be proven to be consistent from within ZFC itself. However, if it is inconsistent, then there will be some statement, such that both the statement itself and its negation are true, and it is certainly possible that you can find proofs of both the statement and its negation.

3. If ZFC is consistent (without us knowing), then we will strictly speaking never know for sure, whether it is consistent or not. However, in an inconsistent system, you can prove that _all_ statements are true statements. My thoughts are that if ZFC was inconsistent after all, then since all statements are true statements, we should by now have come across two proven inconsistent statements, i.e. two inconsistent statements that have both been proven to be true.

Quote:

---

Originally Posted by Mazerakham

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?

---

I presume you mean constructs a counter-example that satisifies the axioms of the real numbers and ZFC. Yes, being able to prove a statement and it contradiction (the counter-exampe) is pretty much the definition of "inconsistent".

Quote:

---

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?

---

No, it has been pretty well proven that the real numbers+ ZFC is NOT inconsistent. And even if it were, that would only mean that those things that specifically depended upon ZFC would be suspect. We know that the axioms for the real numbers [b]are[b] consistent.

Quote:

---

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?

---

Well, I, personally, am not "the mathematical community" but speaking just for myself- no.

Man, that makes me want to find that inconsistency--I'd get pretty good props for that! In the meantime, I just wanted to confirm that I understood this all properly, and I think I do. I'll mark this as "solved," because there's nothing much else to say.

Ah yes, and I'm aware that the set of real numbers can be constructed, and so the consistency of ZFC implies the consistency of the real axioms.

Thanks a lot for the answers.