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 be an infinite cardinal. Then we say that carries a Jónsson algebra if one of the following equivalent conditions hold:
- There is an algebra (in the set-theoretic sense) such that with no proper subalgebras of the same cardinality (such an algebra is called a Jónsson algebra).
- For any (equivalently for some) regular , and any countable expansion of , if is such that and , then .
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 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:
- All measurable cardinals are Jónsson, but the existence of a Jónsson cardinal implies that exists;
- It is equiconsistent with the existence of a measurable cardinal that there is a singular Jónsson cardinal;
- If is a regular Jónsson cardinal, then every stationary subset of reflects;
- If is Jónsson for singular, then is the limit of regular Jónsson cardinals.
I’ve been particularly interested in the consistency of “There exists a singular such that is Jónsson”. In particular, I’ve recently been trying to get a good picture of what the first such 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 -many Jónsson cardinals below . So, getting a good idea of what’s going on here would require me to know what inaccessible Jónsson cardinals below such a 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 is an inaccessible cardinal such that:
- There is a stationary such that does not reflect at inaccessible cardinals.
- There is a sequence such that each is a club subset of , and for every club , there are stationarily-many such that is unbounded in and there is no regular with for all .
- For each , the set of regular Jónsson cardinals below has bounded intersection with .
Then, carries a Jónsson algebra.
What this is telling us is that, if we have a stationary subset of which does not reflect at inaccessibles, and we can find a system of clubs on which guesses clubs in a sufficiently nice way (but off of Jónsson cardinals), then we can find a Jónsson algebra on . So the problem of finding Jónsson algebras actually comes down to building club guessing sequences along particular stationary sets.
Proof: Let be large enough, and let . Let be such that and . In order to show that , we will show that there are arbitrarily large for which . Let be given and set
Note that is a club subset of so we can find stationarily-many such that is unbounded in . Next note that since is a limit cardinal, we know that the set of cardinals below is club in . So we can pick such that:
- is unbounded in ;
- is a cardinal;
The following claim will finish the proof:
Claim: There is some regular such that contains all points of with cofinality greater than .
To see that the claim suffices, note that this allows us to pick some such that:
- carries a Jónsson algebra;
By condition 2. in the hypotheses of the theorem, we can find elements of arbitrarily large cofinality below in there. Then, the claim and assumption 3. allow us to pick an appropriate such . But then, since , we know that is also in as it’s definable from parameters. Further, since , we also get that . This leaves us in the situation that and , and since carries a Jónsson algebra we immediately get that .
Proof of Claim: We have two cases to deal with, of which we will take care of the easier one first.
Case 1 (): This case is relatively easy since the fact that then tells us that . So let , and let be such that , but . This is possible by definition of combined with the fact that is a non-accumulation point of . But then, , that is is the least element of that gets above . So is definable from parameters in and is thus itself in . In this case then, .
Case 2 (): This one is a bit trickier since we don’t have lying around in our model to help us out. Our first job is to find a good candidate for in the claim. With that said, define , which is the least element of which gets above . Note that is a limit ordinal of uncountable cofinality, else we will be able to violate the minimality of (as is not in ). Further, since is a cardinal then must also be a cardinal since and . Thus, since , if is not a cardinal then violates the minimality of . So must be a cardinal of uncountable cofinality.
We now claim that reflects at . That will give us that is a singular cardinal, and thus that . To see that reflects at , note that it suffices to show that this holds in . Otherwise, if there is some club such that , then we will be able to find such a club in by elementarity (as both and are in ). Along those lines, let be club in with . We will show that , so let be given. As , we can find some such that . Further as thinks is unbounded in , we can find such that . On the other hand, the minimality of tells us that and so we’ve found such that . Thus, is an accumulation point of and so is itself in .
At this point, we have shown that must be a singular cardinal. By the minimality of , we then have that . We will now show that every element of with cofinality above is in .
So let be such that , and let be club in with and such that . Note that is not a limit point of , as , so we can find an ordinal satisfying:
Given such an ordinal , we can define the set
Note that since all parameters used to define are in , we have that is in as well. Further, we have that and so is bounded below . Thus, we can find an ordinal satisfying:
Again, we note that all parameters used to define are in and so is also in . Further, recall that we have shown that since is club and . Note that
since both and sit above but below . Thus, . On the other hand, if , we know that , and so in fact . Thus if , then . That is, is the minimum value can take on for . But then is definable from parameters in and so is also in . 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 is a candidate for witnessing that is not Jónsson, then 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 sits in every club such that also appears when showing that every stationary subset of a regular Jónsson cardinal must reflect.