Hang on a minute! Before you include the Axiom of Infinity in your next abstract system, there are two questions you need to ask yourself:
Do you want a system that is simple and powerful or consistent and complete?
Like many Mathematical axioms (best example: Euclid's fifth axiom), interesting things happen in systems where the axiom is denied. In particular, Kurt Gödel showed that systems including the Axiom of Infinity could not be proved consistent or complete. But these properties may be provable in finite systems1.
Do you want a system that is simple or has some relation to reality?
It should be clear that the Axiom of Infinity is empirical in nature. As evidenced by the existential quantifier of predicate logic, allowing abstract systems to make empirical claims can lead to all sorts of ontological problems. Futhermore, an infinite universe may not jibe with your cosmology of choice (see The Universe is Finite).
1: Robert S. Wilson has written a great paper on this idea: http://www.sonoma.edu/People/SWilson/Papers/finite/