A statement referring to an infinite set can sometimes be logically rephrased using only finite sets/objects.  For example, "The set of primes is infinite" <-> "There is no largest prime".  Pleasantly, the proof of this statement does not seem to need infinity either (assume a largest prime, contradiction).  

What reason is there, other than convenience or curiosity, to adjoin infinite sets to our universe by axiomatically declaring that one exists?

Specifically:

> What is an example of a theorem in ZF or ZFC which 1) Does not refer to infinite sets, but 2) Cannot be proven if the Axiom of Infinity is excluded?

(See [Zermelo–Fraenkel set theory][1] for the Axiom of Infinity in context.)


  [1]: http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#The_axioms