# 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| 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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