A travel guide to the large cardinals

Looking at a map of the large cardinal hierarchy for the first time can be a dizzying experience. What are the differences between them, and which ones are similar? Some of them are defined using partition properties and some of them are defined using elementary embeddings, and others have a whole myriad of equivalent characterisations! What’s the intuition about the different sections of the hierarchy, and what type of set theorists are working in each section?

Let’s start out with a bird’s-eye view of the entire enterprise, before we dig into the more fine-grained details. All the diagrams I’ll present here can also be downloaded as pdf’s in my diagrams section.

That’s a lot of large cardinal notions. You’ve probably come across some of them before, maybe even all of them! To get a better understanding of this hierarchy I’ll cut it into four different zones, where the large cardinals in each zone have somewhat similar properties. We start from the bottom.

The combinatorial zone

This section consists of the cardinals starting from the inaccessibles and up to, and including, the weakly compacts. This zone is riddled with various square principles, tree properties and reflection principles, and is therefore the zone combinatorial set theorists tend to lurk. The name is a bit misleading however, as combinatorics permeates most of the large cardinal hierarchy, but with a different flair in terms of colourings and filters.

Note that most of the cardinals are collapsed when we’re looking at them with a consistency lens, which is why we’re mostly interested in direct implications when dealing with this section. I’ve previously blogged about this zone.

The Ramsey-like zone

This zone starts where the combinatorial zone left off, at the weakly compacts, and continues up to the measurables. Most of the cardinals in this area can be characterised as critical points of elementary embeddings between small models of set theory. This zone is also where we start to transcend the constructible universe L, which happens exactly when the small models we’re considering become iterable. We can thus talk about the lower and upper part of this zone. This is the zone I’m dealing with in my recent paper.

The extender zone

Cardinals in this area can be seen as natural strengthenings of the Ramsey-like cardinals, in which we’re still dealing with elementary embeddings between structures, but where we require that the domain of the embeddings is the entire universe V. The simplest such one is the measurable cardinals, and they continue up to the superstrong cardinals.

This could also be called the “inner model theory zone”, as this is the area inner model theorists have mostly been working with. The relatively new area of inner model theory, called descriptive inner model theory, is working in a determinacy hierarchy parallel to the large cardinal hierarchy in this region, which is the right-hand column of the above diagram. I’ve covered this section in a previous blog post.

The crazy zone

I call this the crazy zone for several reasons. Firstly, this is the zone where we reach inconsistency (in ZFC). Secondly, because I don’t have a clue about what’s going on up there. This zone starts with the strongly compacts, and all the cardinals are characterised by embeddings which are usually induced by ultrafilters on $\mathcal P_\kappa(\lambda)$ for some cardinals $\kappa,\lambda$. One notable thing is that combinatorists tend to skip the Ramsey-like and extender zone, and jump straight to the crazy zone, as the strongly compacts and the supercompacts have (surprise surprise) useful compactness properties.

A priori there might be a non-trivial overlap between the extender zone and the crazy zone, but I’m here going by the commonly accepted conjecture that strongly compacts are equiconsistent with supercompacts. To be able to prove this conjecture we need the inner model programme to reach the superstrongs, where the above extender zone diagram shows that it’s still (strictly) below a Woodin limit of Woodin cardinals.