There are two arithmetics of infinity, ordinal & cardinal. I'm going to focus on the cardinal arithmetic as it requires less structure, that is they need less (i.e., ordinals require the idea of ordering whereas cardinals do not).
Cardinal arithmatic recognises that counting can be characterised in two ways, by using numbers, for example 3 means 1+1+1, but there is another way which simply relies on the idea of matching. If I want to see if two bags of beans contains the same number, rather than counting them, I can match them (i.e., one bean from the first matches one bean from the other, and so on), this does not rely on the idea of number and so is more fundamental.
Counting requires number and so can only measure the finite, but matching does without and so go further, and in fact it can measure the infinite, and more it can actually construct an arithmetic. This was first recognised by Cantor one of the inventors of Set Theory.
Does this idea of infinite have any philosophical significance?
I know that Badiou uses Set Theory in his attempt to ground continental philosophy away from post-modern excess (not that I understand how he does it), and he does use the idea of the infinite in this way in his book Being & Event.
Are there any other examples?
Are there arguments against seeing any real significance in this idea of the infinite. For example, Hegel talks of the absolute, and my impression is that in this context, the set theory infinite has no purchase on it.