Infinity Is Weird: What Does It All Mean?

Original

Over at Skull in the Stars [via Jennifer Ouellette]:

The final installment in a series of posts on the size of the infinite, as described in mathematical set theory. The first post can be read here, the second here, and the thirdhere.

We have taken a long, strange journey into the properties of infinity. Over the course of three posts, we have seen that we can characterize the different “sizes” of infinity, though not in the way one might think. We have found, in fact, that there are an infinity of infinities! The smallest one we looked at was the infinite set of counting numbers (labeled $aleph_0$ ); the next largest we found was the continuum (labeled $mathcal{C}$ ): the set of real numbers between 0 and 1. We then found that, for any size infinity, we can construct a larger one.

This leads to an intriguing notion: if we arrange the different size infinities we have found in order, we might have a set of the form

$infty_0=aleph_0, infty_1=mathcal{C}, infty_2, infty_3, ldots$

This would seem to suggest a really elegant possibility: if these are all the infinities, then we could imagine that the set of all infinities form a countable infinity themselves, of size $aleph_0$ , and then we could build up the larger infinities again from this, continuing an endless cycle! For instance, the set of all subsets of the set of all infinities would then be of size $mathcal{C}$ , and so on.

For this to be true, however, we need to know whether there are any other infinities between those we have been able to derive so far. We have shown that there are an infinite number of infinities, but we have not shown that these are the only infinities.

More here.