What are some applications of large cardinals?

Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger cardinals? In a similar vein, can anyone give examples of types of mathematical objects (groups, fields, etc.) that have cardinalities larger than the continuum?

• Are you asking about large cardinals (a specific term - see en.wikipedia.org/wiki/Large_cardinal), or just cardinals which are big? – Noah Schweber Apr 23 '16 at 4:33
• Cardinals which are big. Sorry for the confusion-- didn't realize that "large cardinal" had an exact meaning. – Vik78 Apr 23 '16 at 5:14
• If only there was a short explanation somewhere, maybe in the tag excerpt of large-cardinals... Oh wait... – Asaf Karagila Apr 23 '16 at 7:21
• I posted this on my phone, so that didn't pop up. – Vik78 Apr 25 '16 at 22:03

There are two separate questions here:

• What are some "large" naturally occurring (that is, outside logic :P) mathematical objects?

• What are some applications of large cardinals which live outside logic?

These are really separate questions, and the former has beed addressed elsewhere on this network (see e.g. https://mathoverflow.net/questions/66730/algebraic-structures-of-greater-cardinality-than-the-continuum, https://mathoverflow.net/questions/44705/cardinalities-larger-than-the-continuum-in-areas-besides-set-theory, https://mathoverflow.net/questions/35408/naturally-occuring-groups-with-cardinality-greater-than-the-reals, Groups of cardinality greater than the continuum); I'll focus on the latter.

First, there are situations where large cardinals were initially used to prove a theorem, but then later shown to be unnecessary. The earliest example that I'm aware of (in logic, but I think of interest to other mathematicians) is Martin's proof of Borel determinacy from a measurable, which was later converted to a proof just in ZFC. (But see below!)

Another example of this, in abstract algebra: certain basic results about left distributive algebras were originally proved using (very strong) large cardinal assumptions, which were later removed.

However, there are also examples of statements about left distributive algebras which - to the best of current knowledge - require very strong large cardinal axioms. For example, results about the (finite!) left distributive algebras called Laver tables; see e.g. http://spot.colorado.edu/~szendrei/BLAST2010/miller_new.pdf.

Also, looking back to determinacy, Martin's with-the-measurable proof actually gave analytic determinacy, a result for which large cardinals are genuinely needed; so large cardinals helped "point the way" to extensions of the original theorem, and cast light on what's going on in determinacy principles. Extensions of Martin's argument led to proofs of determinacy of larger and larger classes of reals - e.g., projective - from stronger and stronger large cardinal hypotheses. At this point we're getting kind of deep into logic-y things, so let me give a consequence of projective determinacy which is not provable in ZFC - and indeed has large cardinal strength:

Any set you can get by starting with the irrationals, and taking continuous images and complements, is Lebesgue measurable.

Finally in this category I should mention (even if I don't really understand it) the extensive work by Harvey Friedman on statements in finite combinatorics and in analysis which require large cardinals. He has a number of posts on the "Foundations of Mathematics" mailing list about these, and manuscripts on his site; they are extremely difficult to read, and I tend not to agree with his claims that the combinatorial principles he looks at are "perfectly natural" (I believe that's a direct quotation), but the meta-results he gets are very interesting. Reading (some of) his book Boolean Relation Theory is a goal of mine, for a later date.

• I'm not sure, but to me it seems like OP has a different notion of "large cardinals" in mind, namely mere "cardinalities larger than that of the continuum" which in itself is a rather relative property. That isn't to say however, that I didn't enjoy your answer - quite the contrary. – Stefan Mesken Apr 23 '16 at 4:28
• @Stefan Drat, the title says "large" so I didn't notice the "larger" in the body. Good point. I'll leave this up though since (I hope) it's still interesting. – Noah Schweber Apr 23 '16 at 4:32
• It definitely is still interesting-- I didn't realize that "large cardinals" had a precise formal definition. I was just using it relatively. Sorry for the confusion. – Vik78 Apr 23 '16 at 5:14
• Isn't "Any continuous image of the irrationals is Lebesgue measurable" a ZFC fact? Shelah's result says that "Any $\Sigma^1_3$ set of reals is Lebesgue measurable" implies that "$\omega_1$ is inaccessible in $L$". – hot_queen Apr 24 '16 at 20:38
• @hot_queen Yes, that's right; I started to type the right thing and then just blanked. Fixed. – Noah Schweber Apr 24 '16 at 21:26

Some objects whose cardinal is at least $2^c,$ where $c$ is the cardinal of the reals : (1). The set of all real functions. (2). The set of all filters on an infinite set. (3). The dual space $l_{\infty}^*$ of the Banach space $l_{\infty}.$ (4). The maximal compactifications of $N$,of $Q$,and of $R$. (5). The free group on any set $S$ such that $|S|>c.$

One result that comes to mind is that if $X$ is a separable Tychonoff space and $X$ has a closed discrete subspace $Y$ with $|Y|\geq c$ then $X$ is not a normal space. This relies on the facts that the power set of $Y$ has cardinal at least $2^c$ and that the set of continuous $g:X\to R$ has cardinal at most $c$, and that $c<2^c$. Examples: (1). The Niemitzky plane is not normal. (2). $bN$ has a non-normal subspace, where $bN$ is the maximal compactification of $N.$