Comparing infinite numbers Suppose you have 2 infinite numbers, say $A$ and $B$.
$A$ is an element of the hyperreals, so that $A$ is greater than every real number.
$B$ is the size of the set of natural numbers, $\aleph_0$
Does it make sense to compare $A$ and $B$? And if so, how can you compare these kind of numbers?

These are quite different notions, so the answer I suspect most people will give you is "NO" (and I suspect this question might get closed as well). However, I would like to point out that there are a lot of situations in mathematics where seemingly unrelated notions were later brought under the umbrella of some unifying idea. For example, the convex hull of a set and the sigma-algebra generated by a collection of sets, such as was pointed out yesterday: The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$ Also, there are a lot of situations in mathematics where notational similarity or other essentially non-mathematical resemblance has led to the creation of unifying ideas. For example, fractional derivatives or the idea of complex-iterates of a function.
That said, I doubt anything nontrivial will come out of trying to find a unifying idea underlying these two notions of "infinite number". The only thing that occurs to me is that, in some kind of way that is relative to the underlying "universe" each notion is embedded in, we're looking at the smallest extension outside of certain "small numbers".
