# Nonessential use of large cardinals

In Awfully sophisticated proof for simple facts, we are asked for examples of complex proofs of simple results. To quote from the questioner's post, we are asked for proofs that are akin to "nuking mosquitos." In set theory, a natural "nuke" with respect to a certain result is a large cardinal axiom with unnecessarily high consistency strength (i.e. applying to a much stronger collection of axioms than is required to provide a proof of the possibility of the result in question).

A research focus in set theory is a search for large cardinal axioms with the weakest consistency strength that can be used to prove the possibility of a certain result. My question is of an opposing nature:

Can you think of results that can be proven in a different manner by appealing to a large cardinal axiom with unnecessarily large consistency strength?

There are plenty such examples where the proofs become less technical (e.g., using a $\kappa^{++}$-supercompact cardinal $\kappa$ to show that the GCH can fail at a measurable cardinal is much more than is required), but I'm thinking of examples where the original proof was accomplished without such a strong large cardinal hypothesis or any large cardinal hypothesis at all. For example (from my post to the aforementioned question):

Theorem (ZFC + "There exists a supercompact cardinal."): There is no largest cardinal.

Proof: Let $\kappa$ be a supercompact cardinal, and suppose that there were a largest cardinal $\lambda$. Since $\kappa$ is a cardinal, $\lambda \geq \kappa$. By the $\lambda$-supercompactness of $\kappa$, let $j: V \rightarrow M$ be an elementary embedding into an inner model $M$ with critical point $\kappa$ such that $M^{\lambda} \subseteq M$ and $j(\kappa) > \lambda$. By elementarity, $M$ thinks that $j(\lambda) \geq j(\kappa) > \lambda$ is a cardinal. Then since $\lambda$ is the largest cardinal, $j(\lambda)$ must have size $\lambda$ in $V$. But then since $M$ is closed under $\lambda$ sequences, it also thinks that $j(\lambda)$ has size $\lambda$. This contradicts the fact that $M$ thinks that $j(\lambda)$, which is strictly greater than $\lambda$, is a cardinal.

For the people who are unfamiliar with large cardinal embeddings, let me mention that the critical point of an embedding $j$ is the first ordinal $\kappa$ that is moved (i.e., $j(\alpha) = \alpha$ for all $\alpha$ less than the critical point $\kappa$ and $j(\kappa) > \kappa$.) A cardinal $\kappa$ is $\theta$-supercompact if there exists an elementary embedding $j: V \rightarrow M$ into a transitive (proper class) $M$ with critical point $\kappa$ such that $M^{\theta} \subseteq M$ and $j(\kappa) > \theta$. A cardinal is supercompact if it is $\theta$-supercompact for all $\theta$.

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This may not be the sort of thing you had in mind, but here goes anyway: The easiest way to prove Borel determinacy (which is a theorem of ZFC) is to assume there's a measurable cardinal and prove analytic determinacy. (Both results are due to Tony Martin. The proof of analytic determinacy from a measurable cardinal came well before the proof of Borel determinacy in ZFC. The exact consistency strength of analytic determinacy is the existence of sharps of all reals.)

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+1: Regardless of the order, it is always interesting to find out how a large cardinal axiom was used to simplify the proof of an important ZFC theorem. Thank you. –  Jason Dec 16 '10 at 5:42
@Andreas: This is also the best example I could think of. –  Justin Moore Mar 14 '11 at 23:31

I think this example given at Richard Borcherds's blog would qualify, no?

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+1: The ultimate of "nukes": Laver assumes the existence of a rank-into-rank large cardinal to prove a ZFC theorem! I think this shows that we set theorists may have become a little too complacent in the acceptance of the relative consistency of large cardinal axioms with ZFC. We may be in trouble if someone disproves that possibility one day... –  Jason Dec 10 '10 at 1:36
@Jason: Note that Laver's result was first proved using large cardinals. At that time, no ZFC proof was known. But once you gain some confidence that something is true, it usually becomes easier to actually prove it. I am not sure that Laver actually tried to proof his result and then had the idea to use large cardinals. I would guess it was the other way around, namely that he observed some structure that you get from certain large cardinals, and that this structure actually solves a problem about left-distributive algebras. –  Stefan Geschke Dec 10 '10 at 7:19
@Stefan: I knew that Laver's result preceded the theorem of Dehornoy. My point was that our confidence that a statement is probably relatively consistent with ZFC because we can prove it from one of the strongest known large cardinal notions (not known to be inconsistent with ZFC) might be running a little too high. My concern is what becomes of our research assuming the existence of strong large cardinal notions if it all comes crashing down one day. –  Jason Dec 16 '10 at 5:25
@Jason:I completely disagree with this. I think this is a case where the large cardinals led us to discover a proof (and an algebraic structure) that may not have been discovered or appreciated otherwise. We have about 50 years of experience telling us that a rank to rank is a safe hypothesis. Maybe double that for ZFC. And probably the last 50 years is when we really started to deepen our understanding of CON(ZFC), largely as a consequence of studying large cardinals. So why do we view them with such skepticism? To me, Laver's result is one of the most remarkable of modern mathematics. –  Justin Moore Mar 14 '11 at 23:30
The author has deleted the blog post, so it no longer exists. –  Baby Dragon Aug 28 '13 at 0:58

There is a fantastic (and not too well-known) result of Shelah stating that $L({\mathcal P}(\lambda))$ is a model of choice whenever $\lambda$ is a strong limit of uncountable cofinality.

This is a consequence of a more general theorem that can be found in 4.6/6.7 of "Set Theory without choice: not everything on cofinality is possible", Archive for Math Logic 36 (1997) 81-125.

(Understanding this argument is in my "immediate" to-do list. Alas, the list is longer each day.)

Woodin has a nice, short argument when the cofinality of $\lambda$ is a Woodin cardinal, using stationary tower techniques. (I do not think Woodin's argument is published anywhere, though.) It certainly gives you an idea that the result is plausible, and that an analysis of ideals seems to be relevant.

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+1: This is more of the type of result I had in mind: a proof that proceeded the original one using stronger assumptions. I think this pattern is typical: Shelah comes up with an amazing general result, and his work is then clarified by considering a weaker one. –  Jason Dec 10 '10 at 1:36
I read this before as Shelah's result coming first, but I now realize that you didn't write this. Out of curiosity, do you know which one came first? –  Jason Dec 10 '10 at 2:18
@Jason: Shelah's result appeared first (the paper is dated 1997). I am not sure how Woodin heard of it, but he mentions it in his recent manuscript on "suitable extender models". I believe his argument is fairly recent (he showed it to me in October). –  Andres Caicedo Dec 10 '10 at 2:21
@Andres: OK, thanks. I think this may be another trend: Woodin has so many nice results that he doesn't get a chance to publish them all. –  Jason Dec 10 '10 at 2:36