I was visiting my mom, and discovered that I didn't leave my set theory book on the train; I left it at her house. So I've been happily reunited with my old text, and I'm going to get back to a few more posts about the beautiful world of set theory.
When you talk about set theory, you're talking about an extremely abstract notion, one which is capable of representing all sorts of structures: topological spaces, categories, geometries, graphs, functions, relations, and more. And yet, almost every description of set theory plunges straight into the cardinal and ordinal numbers. Why? That's a question that mystified me for quite a long time. Why do we take this beautiful structure, which can do so many things, and immediately jump in to these odd things about infinities?
I didn't find it out until last year, when I wrote my post about zero. That started me doing
some reading about the history of math, and it's in the historical context that the reason for
this is found. If you're interested in this sort of thing, I recommend the book Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, which
is the most complete, detailed history of math in the 19th and 20th centuries that I know about.
During the 19th century, when math really started to take the shape that we're familiar with, one of
the really big questions was: what is math really? There was a lot of debate about it, but the
most common answer was some variation on "the science of measurement": that is, that math was really about
measuring things, and manipulating those measurements. Things like algebra and calculus were considered
important because of the way that they let you compute measurements.
So when mathematicians were trying to get to the roots of things, they were focused on the
idea of measurement. But what is measurement? It comes down to two different kinds of
things: you can measure how big something is, or where something is. All measurements
of any property can ultimately be reduced to either a measurement of a size, or a measurement of a position, or some combination. For example, to measure the velocity of an object in a plane, you've got two things you need to measure. Speed - which is the size or magnitude of the velocity; and direction, which is the angle (or position) within a circle in which the object is moving.
So Cantor, in his invention of set theory, quite naturally formulated it as an abstract way of talking about the ideas of sizes and positions - which results in two basic kinds of numbers: numbers which describe the size of something, and numbers which describe the position of something. Sets provided a totally abstract way of talking about what those two kinds of numbers meant, and why there was a distinction. And so we got the cardinals for size, and the ordinals for position.
The next step is to try to see what the differences between the two kinds of numbers are. Are they really always the same thing? Or is there some essential semantic difference between an ordinal and a cardinal, between a number that measures size, and a number that measures position? When you sit down and
work it out, you do get a difference - the difference between ω0 and ℵ0, and the fact that cardinals are much larger than the ordinals: the ordinals
are all inevitably countable: position is always a countably infinite value.