FORMALISTS
The issue of infinity produces a slight schism between the two leading figures of the formalist school of thought, namely Russel and Hilbert. Russel concerns himself not with the particulers of a proof of infinity, or even the consistency of infinity, and goes so far as to make the Axiom of Infinity an explicit hypothesis in his Principia Mathematica. Hilbert, alternately, is less trusting of the possible existence of the infinite, and requires a proof of consistency for the infinite to be allowed in a number system. To Hilbert, those facts which can be ascertained combinatorically are self-evidently true, and he employs this rationale in evaluating infinity. Depending upon the circumstance, the consistency of an infinite amount can be determined based upon a finite subset of the infinte, e.g. the first ten digits of an infinitely long series.
INTUITIONISTS
The crux of intuitionist thought is that mathematics cannot be boxed into fixed conceptions of "arbitrary finite magnitude," and this outlook manifests itself wholeheartedly in the approach to infinity. The proof for an infinite item (be it number, function, etc.) is one of induction. This involves applying formal proof techniques only to the most basic infinite structure, and subsequently inferring that this holds true for infinite repetition of the event.
Example: the proof of the fact that the sum of the first n odd numbers[(1+3+...+(2n-1))] is a perfect square, n^2, begins with the base step that if
n = 1, then 1=1^2.
Next, the validity of n is shown by using (n+1)^2
=n^2+2n+1
since the n+1 odd number in the sequence is (2n-2)+1, then n^2+2n+1 = n^2 + n+1st in sequence