Starting to understand epsilon-zero

This post is going to be about what infinite ordinal numbers are, and about is in particular. I had a brainwave a while back (18 months now, wow, I have definitely not been blogging enough) and suddenly understood much better than I did before. I have several related ideas here and I am going to try to write one blog post about each of them, instead of one gigantic blog post about all of them together that I never finish. I really like the ordinal numbers. For some reason I was repeatedly exposed to the infinite cardinals as a child and, while they are pleasingly mysterious, they're also somewhat uninteresting because they have no internal structure, they are just bignesses. It's super cool that there is more than one possible bigness of an infinite set, of course, but sets can have all sorts of interesting structure, and looking just at the bigness ignores all that. The ordinals are much more satisfying, and also I feel that they are more like numbers. This post explains how they…