Jónsson Algebras on Inaccessibles

I want to use this post to work through a result of Shelah’s the uses club guessing to produce Jónsson cardinals on inaccessible cardinals. My main reason is that a few of the arguments that appear get used a number of times in Cardinal Arithmetic. First, some definitions and motivation (anyone interested more in the result itself can just skip down):

Lemma/Definition: Let \lambda be an infinite cardinal. Then we say that \lambda carries a Jónsson algebra if one of the following equivalent conditions hold:

  1. There is an algebra (in the set-theoretic sense) \mathfrak{A}=(A, (f_n)_{n<\omega}) such that |A|=\lambda with no proper subalgebras of the same cardinality (such an algebra is called a Jónsson algebra).
  2. For any (equivalently for some) regular \theta\geq\lambda^+, and any countable expansion \mathfrak{A} of (H(\theta),\in,<_\theta), if M\prec \mathfrak{A} is such that \lambda\in M and |M\cap\lambda|=\lambda, then \lambda\subseteq M.
  3. \lambda\not\rightarrow[\lambda]^{<\omega}_\lambda.

The proof that these three conditions are equivalent is pretty standard, and can be found in a number of places (Kanamori’s The Higher Infinite being one of them).

We say that a cardinal \lambda is Jónsson if it carries no Jónsson algebras. It turns out that these things are interesting enough that there are three chapters in Cardinal Arithmetic with the purpose of clarifying the situation at Jónsson cardinals. Here are some quick facts:

  1. All measurable cardinals are Jónsson, but the existence of a Jónsson cardinal implies that 0^\sharp exists;
  2. It is equiconsistent with the existence of a measurable cardinal that there is a singular Jónsson cardinal;
  3. If \lambda is a regular Jónsson cardinal, then every stationary subset of \lambda reflects;
  4. If \lambda=\mu^+ is Jónsson for \mu singular, then \mu is the limit of regular Jónsson cardinals.

I’ve been particularly interested in the consistency of “There exists a singular \mu such that \mu^+ is Jónsson”. In particular, I’ve recently been trying to get a good picture of what the first such \mu should look like. For example, we know that it has to be countable, and a limit of inaccessible Jónsson cardinals. On the other hand, I have no idea whether or not there actually needs to be \mu-many Jónsson cardinals below \mu. So, getting a good idea of what’s going on here would require me to know what inaccessible Jónsson cardinals below such a \mu look like. Luckily, Shelah’s devoted a lot of work in Cardinal Arithmetic to this kind of stuff.

I want to start with a result that appears relatively early on in chapter 3 of Cardinal Arithmetic. There is an analogous result for successors of singular cardinals, and in fact Shelah proves both results at the same time. Unfortunately, while the proofs do follow a similar pattern, they are not similar enough to warrant this treatment. So, I decided to tease out the part about inaccessible cardinals.

Theorem (Shelah): Suppose that \lambda is an inaccessible cardinal such that:

  1. There is a stationary S\subseteq \lambda such that S does not reflect at inaccessible cardinals.
  2. There is a sequence \bar C=\langle C_\delta : \delta\in S\rangle such that each C_\delta is a club subset of \delta, and for every club E\subseteq \lambda, there are stationarily-many \delta\in S such that E\cap nacc (C_\delta) is unbounded in \delta and there is no regular \gamma<|\delta| with cf(\alpha)<\gamma for all \alpha\in E\cap nacc(C_\delta).
  3. For each \delta\in S, the set of regular Jónsson cardinals below \delta has bounded intersection with nacc (C_\delta).

Then, \lambda carries a Jónsson algebra.

What this is telling us is that, if we have a stationary subset of \lambda which does not reflect at inaccessibles, and we can find a system of clubs on S which guesses clubs in a sufficiently nice way (but off of Jónsson cardinals), then we can find a Jónsson algebra on \lambda. So the problem of finding Jónsson algebras actually comes down to building club guessing sequences along particular stationary sets.

Proof: Let \theta\geq \lambda^+ be large enough, and let \mathfrak{A}=(H(\theta),\in,<_\theta, S,\bar C). Let M\prec \mathfrak{A} be such that |M\cap \lambda|=\lambda and \lambda\in M. In order to show that \lambda\subseteq M, we will show that there are arbitrarily large \sigma<\lambda for which \sigma\subseteq M. Let \sigma<\lambda be given and set

E=\{\alpha<\lambda : \sup(\alpha\cap M)=\alpha\}.

Note that E is a club subset of \lambda so we can find stationarily-many \delta\in S\cap E such that E\cap nacc(C_\delta) is unbounded in \delta. Next note that since \lambda is a limit cardinal, we know that the set of cardinals below \lambda is club in \lambda. So we can pick \delta such that:

  • \delta\in S\cap E;
  • E\cap nacc(C_\delta) is unbounded in \delta;
  • \delta is a cardinal;
  • \delta>\sigma.

The following claim will finish the proof:

Claim: There is some regular \kappa<\delta such that M contains all points of E\cap nacc(C_\delta) with cofinality greater than \kappa.

To see that the claim suffices, note that this allows us to pick some \alpha\in nacc(C_\delta)\cap E) such that:

  • cf(\alpha)>\sigma;
  • cf(\alpha) carries a Jónsson algebra;
  • \alpha\in M.

By condition 2. in the hypotheses of the theorem, we can find elements of arbitrarily large cofinality below \delta in there. Then, the claim and assumption 3. allow us to pick an appropriate such \alpha. But then, since \alpha\in M, we know that cf(\alpha) is also in M as it’s definable from parameters. Further, since \sup M\cap \alpha=\alpha, we also get that \sup M\cap cf(\alpha)=cf(\alpha). This leaves us in the situation that |M\cap cf(\alpha)|=cf(\alpha) and cf(\alpha)\in M, and since cf(\alpha) carries a Jónsson algebra we immediately get that \sigma\subseteq cf(\alpha)\subseteq M.

Proof of Claim: We have two cases to deal with, of which we will take care of the easier one first.

Case 1 (\delta\in M): This case is relatively easy since the fact that \delta, S, \bar C\in M then tells us that C_\delta\in M. So let \beta\in nacc(C_\delta)\cap E, and let \beta^*\in M be such that \beta^*>\sup (C_\delta\cap\beta), but \beta^*<\beta. This is possible by definition of E combined with the fact that \beta is a non-accumulation point of C_\delta. But then, \beta=\min\{\alpha\in C_\delta : \alpha>\beta^*\}, that is \beta is the least element of C_\delta that gets above \beta^*. So \beta is definable from parameters in M and is thus itself in M. In this case then, E\cap  nacc(C_\delta)\subseteq M.

Case 2 (\delta\notin M): This one is a bit trickier since we don’t have C_\delta lying around in our model to help us out. Our first job is to find a good candidate for \kappa in the claim. With that said, define \beta_\delta=\min(M\cap\lambda\setminus\delta), which is the least element of M\cap \lambda which gets above \delta. Note that \beta_\delta is a limit ordinal of uncountable cofinality, else we will be able to violate the minimality of \beta (as \delta is not in M). Further, since \delta is a cardinal then \beta_\delta must also be a cardinal since |\beta_\delta|\geq\delta and \delta\notin M. Thus, since |\beta_\delta|\in M, if \beta_\delta is not a cardinal then \beta_\delta>|\beta_\delta|\geq \delta violates the minimality of \beta_\delta. So \beta_\delta must be a cardinal of uncountable cofinality.

We now claim that S reflects at \beta_\delta. That will give us that \beta_\delta is a singular cardinal, and thus that cf(\beta_\delta)<\delta. To see that S reflects at \beta_\delta, note that it suffices to show that this holds in M. Otherwise, if there is some club d\subseteq \beta_\delta such that S\cap d=\emptyset, then we will be able to find such a club in M by elementarity (as both \beta_\delta and S are in M). Along those lines, let d\subseteq\beta_\delta be club in \beta_\delta with d\in M. We will show that \delta\in acc(d), so let \alpha<\delta be given. As \delta\in E, we can find some \beta\in M such that \alpha<\beta<\delta. Further as M thinks d is unbounded in \beta_\delta, we can find \gamma\in d\cap M such that \gamma>\beta. On the other hand, the minimality of \beta_\delta tells us that \gamma<\delta and so we’ve found \gamma\in d such that \alpha<\gamma<\delta. Thus, \delta is an accumulation point of d and so is itself in d.

At this point, we have shown that \beta_\delta must be a singular cardinal. By the minimality of \beta_\delta, we then have that cf(\beta_\delta)<\delta. We will now show that every element of E\cap nacc (C_\delta) with cofinality above cf(\beta_\delta) is in M.

So let \beta\in E\cap nacc(C_\delta) be such that cf(\beta)>cf(\beta_\delta), and let d\subseteq \beta_\delta be club in \beta_\delta with d\in M and such that |d|=cf(\beta_\delta). Note that \beta is not a limit point of d, as cf(\beta)>|d|, so we can find an ordinal \beta_0 satisfying:

  • \beta_0\in M;
  • sup(C_\delta\cap\beta)<\beta_0<\beta;
  • sup (d\cap\beta)<\beta_0.

Given such an ordinal \beta_0, we can define the set

A=\{\min (C_\epsilon\setminus\beta_0) : \epsilon\in d\cap S\}.

Note that since all parameters used to define A are in M, we have that A is in M as well. Further, we have that |A|\leq |d|<cf(\beta) and so A\cap\beta is bounded below \beta. Thus, we can find an ordinal \beta_1 satisfying:

  • \beta_1\in M;
  • \beta_0<\beta_1<\beta;
  • A\cap [\beta_1,\beta)=\emptyset.

Now define

d^*=\{\epsilon\in d\cap S\setminus\beta_1 : \min(C_\epsilon\setminus\beta_0)=\min(C_\epsilon\setminus\beta_1)\}.

Again, we note that all parameters used to define d^* are in M and so d^* is also in M. Further, recall that we have shown that \delta\in S\cap d since d\subseteq\beta_\delta is club and d\in M. Note that

\min(C_\delta\setminus\beta_0)=\beta=\min (C_\delta\setminus\beta_1),

since both \beta_0 and \beta_1 sit above \sup (C_\delta\cap\beta) but below \beta. Thus, \delta\in d^*. On the other hand, if \min(C_\epsilon\setminus\beta_0)<\beta, we know that A\cap [\beta_1,\beta)=\emptyset, and so in fact \min(C_\epsilon\setminus\beta_0)<\beta_1. Thus if \epsilon\in d^*, then \min(C_\epsilon\setminus\beta_0)\geq\beta. That is, \beta is the minimum value \min(C_\epsilon\setminus\beta_0) can take on for \epsilon\in d^*. But then \beta is definable from parameters in M and so is also in M. That completes the proof of the claim, as well as the theorem.

So one cool thing about the proof of this theorem is that most of the arguments here appear frequently when relating Jónsson algebras to club guessing. The primary vehicle this relation is actually the claim which ended up being the major crux of the proof. Basically, it allows us to conclude that, if M is a candidate for witnessing that \lambda is not Jónsson, then M will be able to see enough of the ladder systems which guess clubs. So the ladder systems that are guessing clubs (as opposed to the clubs that are guessed themselves) end up being the vehicle for moving up Jónssonness. Interestingly enough, the sets along which scales appear also serve a similar purpose when considering successors os singular cardinals. Additionally, our argument showing that \delta sits in every club d\subseteq\beta_\delta such that d\in M also appears when showing that every stationary subset of a regular Jónsson cardinal must reflect.

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