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 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 where 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 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 , or else my argument would have been circular. To do this, one proceeds as follows. If is a morphism in Comp X, then since X is dense in Z, the underlying continuous surjection is completely determined by its behavior on X, but it is also the identity on X. Therefore Comp X is a poset category. Let be a chain in Comp X; then is an inverse system of topological spaces, and if C is the inverse limit of , then one can show that there is a closed embedding . Since is a compact Hausdorff space by Tychonoff’s theorem, so is C. Taking the inverse limits of the open dense embeddings , where , we obtain an open dense embedding , so C is an upper bound of 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 is a large category. To see that Comp X is equivalent to a small category, it suffices to show that there is a cardinal such that every compactification of Comp X has at most points; then for every compactification Y of X, one can find a compactification Z of X such that in Comp X, and the set-theoretic rank of Z is at most , and so Comp X is a subset of the set . Furthermore, if Y is a compactification of X and , 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 . 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 is the cardinality of X, then there are only 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 . Therefore we may let .

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

Definitions. Let be a regular cardinal. We say that is an inaccessible cardinal if for every cardinal , . We say that is a hyperinaccessible cardinal if is an inaccessible cardinal and there is an increasing chain of inaccessible cardinals such that .

Let be a hyperinaccessible cardinal and suppose that . Then there are inaccessible cardinals . If and Y is a compactification of X, then Y can be obtained as an extension of the Alexandroff compactification by splitting nets, but is a Grothendieck universe and so the topology of X can be already probed by nets in ; therefore . Therefore is a small category in , so X has a Stone–Čech compactification with .

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 , then enough nets to probe the topology of X are also under , so every compactification is as well.