I reckon the reasoning behind of some axioms in set theory. The axiom of extent is to say that we only care about what the sets contain, an abstraction for containers; the axiom of pair, the axioms of union, the axiom of power make sense for the construction of sets, from others; since the axiom of subsets is impossible to be proven by the others axioms, as I unsurely believe, then its assertion is essential. The axiom of regularity makes the felling of stupidity emanate from the reader, for its wonder.
But what in the hell is with the axiom of infinity? It seems like it only exists to say that the set of the natural numbers exists, so why it just does not assert that? My only is guess is (optional read):
that it is as if the axiom of infinity asserted infinity, and the way it does that we use to build the natural numbers; but, whilst making those axioms, the intention was for the existence of the natural numbers and the existence of infinity to be related, since the set of the natural numbers looks like the simplest infinity, and then we build from that, other infinity-things. That is, we assert the existence of infinity by asserting the existence of the simplest kind of infinity. So the axiom of infinity does not simply declare the existence of the set of the natural numbers because it has bigger plans: it asserts the existence of infinity using the natural numbers (asserting its existence), but it is not FOR the natural numbers (for asserting its existence).
Why is the axiom of infinity defined as it is, and has it any use besides proving the existence of the set of the natural numbers?
Best Answer
In ZFC with the axiom of infinity excluded you can build sets of arbitrary size (you can make do with the axioms of the empty set and pairing), however the cardinality of the resulting set will always be finite. The axiom of infinity essentially asserts that there is a set that is one single object that is infinite in size. And yes, this means the smallest kind of infinity, "on top of which" other infinities (e.g. the continuum) are "built".
This is a very short answer, there is still a lot to say about the axiom, both from mathematical and philosophical points of view.