One of the most interesting topics in the philosophy of mathematics is the Axiom of Choice. Here is a nice page with a little intro to the topic and also lots of links for further reading.
For the philosophical relevancy, this quote is a nice demonstration:
Jerry Bona once said,
The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn’s Lemma?
This is a joke. In the setting of ordinary set theory, all three of those principles are mathematically equivalent — i.e., if we assume any one of those principles, we can use it to prove the other two. However, human intuition does not always follow what is mathematically correct. The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians; and Zorn’s Lemma is so complicated that most mathematicians are not able to form any intuitive opinion about it.
Also be sure to read a bit about the Banach-Tarski Paradox, which shows that AC can lead to problems when applying theories which use AC to problems of physics. (Of course, this is not a practical problem because to my knowledge most (all?) of the math used for physics can also be expressed without AC, although in a more complicated exposition.)
But one conclusion can be drawn: one must be very careful in introducing (uncountable) infinities an also when reasoning with infinities. Our intuitions have evolved and are learned in a finite, albeit vast world (also in space, we can’t see past the Hubble volume).
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