I was idly parsing the blog on XKCD’s site, and came across this post concerning large numbers. Reading it led me to this page, which is quite an excellent summation of how cool really big numbers are (in terms a non-mathematician can more or less understand) and also gives a pretty good basic introduction to the concepts of cardinal and ordinal infinities.
My favorite paragraph:
At this point in the discussion (or usually sooner) it becomes apparent that there is additional knowledge and assumptions “outside the system”. An effort is made to identify these, define them precisely and add them into the quantity N. After doing this, it is soon discovered that the resulting formal system itself depends on things outside itself, and so on. I have encountered many expositions, discussion threads, etc. over the years, that begin with an optimistic determination to formalize the problem and quantify exactly how large numbers can be derived from first principles; they all have ended up somewhere in this jungle of abstraction. Here is a relevant quote:
I have this vision of hoards of shadowy numbers lurking out there in the dark, beyond the small sphere of light cast by the candle of reason. They are whispering to each other; plotting who knows what. Perhaps they don’t like us very much for capturing their smaller brethren with our minds. Or perhaps they just live uniquely numberish lifestyles, out there beyond our ken.