I’m currently working my way through portions of Chapter III and IV of Cardinal Arithmetic to get a hold on some of the material regarding Jónsson algebras on inaccessibles. For now though, I want to make a short post which talks about the club guessing ideals which feature prominently in these chapters. In particular, I want to use the statement of Claim 1.9 from Chapter III of Cardinal Arithmetic to motivate the definitions of these ideals. Let’s start by recalling the statement:

**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.

The statement of the theorem is a bit of a mess, but that’s partially my fault for trying to use as little notation as possible. So, let’s start with the notation.

**Definition**: Let be stationary. We say that is an -club system if each is a club subset of .

The statement of the above theorem is about particular types of -club systems. In particular, we’re asking that anticipate clubs stationarily-often in the following sense:

given any club , there are stationarily-many such that is “not too small”.

This particular notion of not being too small leads us to associate to our -club system , a system of ideals as follows.

Each is an ideal over generated by the following sets:

- ;
- for each ;
- for each regular ;

Then we see that the manner in which we want to guess clubs is that we need stationarily-often for any club . So let’s say that the pair guesses clubs if this happens. Then the statement of the theorem becomes:

**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 the pair guesses clubs.
- For each , the set of regular Jónsson cardinals below is in .

Then is not Jónsson.

I want to point out that we could ask about guessing clubs relative to other sequences of ideals, and Shelah does precisely that. The ideals we’ve concocted however, are the ones most often considered as they seem to be the ones most directly relevant to the problem of producing Jónsson algebras. Now the next thing to note is that the crux of the proof was the fact that the elementary submodel inconsideration sees enough points of where was a particular club subset of . So we didn’t need the full force of condition 3. If we let we only needed to produce a club guessing sequence such that:

For every club , there are stationarily-many such that .

In other words, we are asking that guesses clubs. This leads us to yet another ideal that features prominently in the literature, and captures precisely when this doesn’t happen.

**Definition**: Let be stationary, be an -club system, and let be a sequence of ideals (with an ideal on ). We define the ideal by saying that if and only if there is some club such that for every , we have .

This allows us to state yet another version of the theorem (if we let be defined as before):

**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 the pair guesses clubs.
- The set of regular cardinals which carry a Jónsson algebra is not in .

Then is not Jónsson.

In the next entry, I want to (using the above theorem) give a proof that any inaccessible Jónsson cardinal must either be an inaccessible limit of Jónsson cardinals or a Mahlo cardinal. This is not near the state of the art, but it’s a good place to start, as the proof involves showing how to produce the sorts of club guessing sequences that we need.