Infinite Sets (1st yr Uni) and Axioms This is my first question, so please be easy on me if it's too "conversational".
I've been asked to consider how the axioms of ZFC Set Theory, and any that underlie it, could influence the way in which mathematicians approach the subject of infinite sets. This is for a discussion (there is no "answer"), but I think I've found a fundamental issue with a proof that underpins modern mathematics.
I don't want to look like a fool at University, but I can't get the idea out of my head (surely I must be wrong... it has been a century since it was settled).
My question is:
What is the best way to verify a hypothesis if you lack a full understanding of the subject area?
Thanks.
 A: To be honest, the only options are (1) to acquire a full understanding of the subject area, or (2) to ask someone who already has it. A major issue here is that mathematics isn't an experimental science, and math having to do with infinity is especially not - so you can't just "test" the hypothesis in your basement.
Judging from the comments, you're concerned that an expert might be too "close" to the topic and become unable to see the problem; that's not something to worry about here. First of all, theorems like Cantor's Diagonalization were not always accepted - when first introduced, Cantor's theorems were heavily examined by experts who were predisposed to believe they were false. Second of all, math has the virtue that a proof is either correct or not - there's no opinion to be had there. Possibly you've noticed a problem that an expert wouldn't have seen, but if you point it out to them an expert will be able to tell whether it's really a problem. And if it isn't, they'll be able to explain to you what you're misunderstanding.
The natural next question is: where do you find such an expert? One option might be to go to your professor, but you've expressed a fear of "looking like a fool" - so I'd recommend someone you don't know, so that you don't care what they think of you. A ready source of such experts is right here, on StackExchange! You could just post your proposed flaw in the proof as a question, and you'll probably get a response very quickly.
One last note: I would consider it honestly very unlikely that what you've found is actually a flaw in the proof. The result that there are different sizes of infinity was a deeply surprising one, and as a result Cantor's theorems are among the most thoroughly cross-examined theorems in mathematics. However, I would encourage you to talk to someone about your idea - even if you aren't correct in thinking that the proofs fail, it would be useful for you as a scholar to find out why they don't.
