Much ado about large cardinals

Lately, with Peter Scholze’s MathOverflow post about Grothendieck universes and the Isabelle/HOL implementation of schemes, it seems that in the sphere of online math there has been a somewhat renewed interest in when large cardinals make proving theorems easier. (Specifically, it is not necessary that one actually needs the large cardinals to prove the theorem — only that it makes the proof easier!) So I thought it would be fun to look through some old homework of mine and see if I could find an example where if I had allowed myself the use of a large cardinal, my life would have been easier. I found an example from when I took a course in C*-algebras a few years ago.

Let X be a locally compact Hausdorff space. By a compactification of X we mean an open dense embedding X \to Y where Y is a compact Hausdorff space. By Alexandroff’s theorem, X always has a compactification, but in general if X is not compact then X may have multiple compactifications. We consider the category Comp X of compactifications of X equipped with continuous surjections which preserve X; the Alexandroff compactification is the final object of Comp X.

The Stone–Čech theorem. The category Comp X has an initial object.

One may show that the initial object of Comp X is \text{Spec } C_b(X) where C_b(X) is the Banach space of bounded continuous functions on X with its supremum norm, and the functor Spec is taken in the sense of C*-algebras (thus Spec A consists of maximal closed ideals equipped with the Zariski-Jacobson topology). This proof is presumably inoffensive to anyone who accepts ZFC (and offensive to anyone who does not, since one needs Zorn’s lemma to show that C_b(X) has a maximal ideal in general — and ZF alone cannot prove that Comp X has an initial object).

However, for the purposes of the result I was trying to prove, I needed a proof of the Stone–Čech theorem that did not rely on the existence of \text{Spec } C_b(X), or else my argument would have been circular. To do this, one proceeds as follows. If Z \to Y is a morphism in Comp X, then since X is dense in Z, the underlying continuous surjection Z \to Y is completely determined by its behavior on X, but it is also the identity on X. Therefore Comp X is a poset category. Let \mathcal C be a chain in Comp X; then \mathcal C is an inverse system of topological spaces, and if C is the inverse limit of \mathcal C, then one can show that there is a closed embedding C \to \prod \mathcal C. Since \prod \mathcal C is a compact Hausdorff space by Tychonoff’s theorem, so is C. Taking the inverse limits of the open dense embeddings X \to Y, where Y \in \mathcal C, we obtain an open dense embedding X \to C, so C is an upper bound of \mathcal C in Comp X.

At this point, one may proceed in two ways. Working in ZFC, it is only valid to apply Zorn’s lemma if Comp X is equivalent to a small category, but \text{Comp } \mathbb N is a large category. To see that Comp X is equivalent to a small category, it suffices to show that there is a cardinal \kappa such that every compactification of Comp X has at most \kappa points; then for every compactification Y of X, one can find a compactification Z of X such that Y \cong Z in Comp X, and the set-theoretic rank of Z is at most \kappa, and so Comp X is a subset of the set V_\kappa. Furthermore, if Y is a compactification of X and y \in Y, then, since X is dense in Y, by the boolean prime ideal theorem there is an ultrafilter U on the set Open X of open subsets of X such that \lim U = y. Since Y is Hausdorff, it follows that y is the UNIQUE limit of U, but some cardinal arithmetic can be used to show that if \lambda is the cardinality of X, then there are only 2^{2^\lambda} ultrafilters on Open X (since elements of an ultrafilter on Open X are open subsets of X), so the cardinality of Y is at most 2^{2^\lambda}. Therefore we may let \kappa = 2^{2^{\lambda}}.

Okay, that was stupid. We can also proceed by large cardinals. The following argument feels much more conceptual to me:

Definitions. Let \delta > \aleph_0 be a regular cardinal. We say that \delta is an inaccessible cardinal if for every cardinal \lambda < \delta, 2^\lambda < \delta. We say that \delta is a hyperinaccessible cardinal if \delta is an inaccessible cardinal and there is an increasing chain of inaccessible cardinals \delta_\alpha such that \lim_\alpha \delta_\alpha = \delta.

Let \delta be a hyperinaccessible cardinal and suppose that \text{card }X < \delta. Then there are inaccessible cardinals \text{card }X < \kappa < \kappa' < \delta. If X \in V_\kappa and Y is a compactification of X, then Y can be obtained as an extension of the Alexandroff compactification by splitting nets, but V_\kappa is a Grothendieck universe and so the topology of X can be already probed by nets in V_\kappa; therefore Y \in V_\kappa. Therefore \text{Comp } X \subseteq V_\kappa is a small category in V_{\kappa'}, so X has a Stone–Čech compactification \beta X with \text{card } \beta X < \kappa' < \delta.

This argument looks verbose, but only because I have written out the details; I think in practice I would just say that if X lies underneath an inaccessible cardinal \kappa, then enough nets to probe the topology of X are also under \kappa, so every compactification is as well.


3 thoughts on “Much ado about large cardinals

  1. Okay thanks. I thought I had read something incorrect about Stone-Cech compactifications in the past. This clarifies things.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s