Thanks to a good friend of mine, I’m finally learning set theory. I’m going through a textbook; he checks my work and helps me out if I’m not following.
Set theory seems, at first glance, to be incredibly simple. Sets are just collections of other sets. So, if I have set A and set B, I might then be able to create set C, the set that has A and B in it. Or I can create set D, the set that has only set A in it. Or set E, the set that has set A and another set – call it F – which in turn has B and C in it. And so, from this simple operation of “set membership”, one can create systems so complex that all of arithmetic can be recreated in terms of sets.
You might ask – how did I get sets A and B in the first place? If sets only have other sets in them, do we have a “starting set” from which to build? Indeed, we do: the empty set. The empty set has no sets in it. But we can create a set of the empty set. That has one set in it – namely, the empty set. And we can create the set that has the empty set and the set of the empty set. And off we go.
All of this means that the gargantuan, entangled edifice that is mathematics is built on a single object – which we might poetically call nothing – and a single basic operation: “is a member of”.…