How can one best visualize a measurable cardinal?

Question

I've searched the web for several years attempting to find an intuitive and visualizable explanation of what measurable cardinals are and how best to conceive of them.

All I've found are definitions. For example: a two-valued measure on a cardinal $\kappa$, or more generally, on any set. A cardinal, $\kappa$, can be described as a subdivision of all of its subsets into large and small sets such that

1. $\kappa$ itself is large
2. $\varnothing$ and all singletons $\{\alpha\}, \alpha \in \kappa$ are small
3. Complements of small sets are large and vice versa
4. The intersection of fewer than $\kappa$ large sets is again large​.

I know what these various terms mean, but that doesn't help me:

a. Visualize what a measurable cardinal looks like

b. Know how one makes a measurable cardinal

c. Understand how Ulam first "discovered" -- or "invented"? -- them by working on the Lebesgue measure problem troubling everyone in the late 1920s

d. Know whether they can all still be reached by some Beth cardinal, say Beth omega, or Beth c -- presumably not!

e. Know why they have the amazing properties they do including that of being vastly larger than all of the inaccessible cardinals below them

Answer 1

In mathematics you don't understand things, you get used to them, to quote von Neumann.

Measurable cardinals, and really any large cardinal property, are definitions that are nearly impossible to visualize in any meaningful way. There are two reasons for this:

1. We are not very good at visualizing sets without structure, so we are not very good at visualizing the set theoretic universe, and the membership relation between the set. Not to mention that it gets very complicated, very fast, making it nearly impossible to fully appreciate.

2. Large cardinals are huge in comparison to sets that you normally think of. So large, that however large you think they might be, they are probably much larger. Every first-order definition which you could think of describing an inaccessible cardinal has already occurred at a lot of cardinals below it. Every second-order definition you might want to consider giving you a measurable cardinal has given you a lot of cardinals below it satisfying this definition.

   In particular, you need third-order logic to truly say that $\kappa$ is a measurable cardinal in the structure $V_\kappa$ (and that makes sense, as a measure is an object which lies in $V_{\kappa+2}$).

So how can you understand it? Well. You have to use measurable cardinals for a while, and get comfortable with them.

Just like understanding integrals as the area or volume of something is helpful intuition, but ultimately, you can integrate functions that have no reasonable "visual representation", so what sense could your brain make of volume and area they graph encloses? Well. You have a definition, and you work with it, and you got used to it, and everything is fine. Same with continuity, or determinants, or partitions of a set, or an arbitrary function being injective.

At first the definition was cumbersome, and you didn't quite get it. Maybe you made some mistakes, and maybe you didn't. But you worked slowly with the definitions and theorems until you got the feel of what these things are used for, and then you developed an image in your head.

So what about measurable cardinals? Well. There is a definition, which you cite. There is a free $\kappa$-complete ultrafilter on $\kappa$. That's it. Then you can start using this for actually proving things, like $V\neq L$, or the generic absoluteness of $\Sigma^1_3$ sentences, and after you've proved a bunch of things using measurable cardinals, you get better at understanding what they might be. But you can't quite give this intuition away to people. The most you can do is explain what these cardinals are useful for, and try and guide the development of that intuition in other people. (And again, this is true for everything in mathematics, not just large cardinals or set theory.)

Answer 2

Let me address (e), (d), (b) in that order.

(e) is wrong: measurables are not greater than every strongly inaccessible cardinal. (For example, every measurable is itself strongly inaccessible!) A measurable is, however, greater than lots of strongly inaccessible cardinals - in fact, if $\mu$ is measurable then there are $\mu$-many strongly inaccessible cardinals $<\mu$.

I'm not sure what (d) means. Every cardinal is an $\aleph$-number (assuming choice), and in particular $\mu=\aleph_\mu$ for every measurable $\mu$. But that by itself isn't very mysterious - there are rather small cardinals satisfying $\kappa=\aleph_\kappa$.

For (b), the way one gets a measurable cardinal is (usually) via an elementary embedding of the universe into (some part of) itself. This is a bit technical, but I think it's worth stating, and either Jech or Kanamori can fill in the details and provide more information:

If there is some class $M\subset V$ and some map $j:V\rightarrow M$ which is a nontrivial elementary embedding of $(V,\in)$ into $(M, \in\upharpoonright M)$, then (easy exercise) there is some least ordinal $\kappa=crit(j)$ such that $j(\kappa)>\kappa$. This turns out to be a measurable cardinal - specifically, the set $$\{X\subseteq \kappa: \kappa\in j(X)\}$$ is a measure on $\kappa$ for $\kappa=crit(j)$ (hard exercise). So one way to say "there is a measurable cardinal" is "there is a nontrivial elementary embedding of the universe into some inner model." And this in fact is how many higher large cardinal properties are usually stated and motivated: as asserting the existence of an embedding, or family of embeddings, with certain properties (where the large cardinal in question is (usually) the critical point of the embedding(s)). (Interestingly, there can be no proper elementary embedding $V\rightarrow V$, so there is a limit to how far this idea can be pushed before it explodes.)