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”.
I’m simplifying, but only slightly. There are various axioms to set theory that tell us how we may construct sets out of already existing ones. And those axioms aren’t quite as simple as they might be, thanks to our good friend Russel’s paradox. But I think it’s fair to say that the broad insight stands.
This is all rather reminiscent of the Buddhist image of Indra’s net: the network of reflections of reflections of reflections. Imagine a net: it has nodes, and lines connecting the nodes. Each node – typically pictured as pearls – reflects the entire net (except iself). But now forget the pearl, and imagine each node as nothing more than the reflection of all the other nodes. What we’re left with is a net, each node of which is the reflection of the entire net (from its point of view, so to speak). And within each reflection, only reflections are reflected. So it’s reflections all the way down.
A single object – a node – and a single operation – reflection.
The analogy isn’t quite perfect. For one, all nodes contain the reflection of all other nodes in Indra’s net, whereas this isn’t the case with sets. This is because, although both set theory and Indra’s net are built off emptiness, they do so quite differently. Set theory works hierarchically, taking emptiness as its bottom and building up from it. Thus, every step in the building process further complicates the edifice. Indra’s net, on the other hand, doesn’t have a bottom. It is emptiness through and through, in all directions. Every iteration is already infinitely complex.
I’m out of my element at this point. There is a lot more for me to learn about set theory – I am sure this analogy will either deepen or break down once I learn more about, for example, infinite sets, and non-well-founded sets. But I’m rather fond of my beginner’s observation.