Club Guessing Ideals

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 \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.

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 S\subseteq \lambda be stationary. We say that \bar C=\langle C_\delta : \delta \in S\rangle is an S-club system if each C_\delta is a club subset of \delta.

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

given any club E\subseteq \lambda, there are stationarily-many \delta\in S such that nacc(C_\delta)\cap E is “not too small”.

This particular notion of not being too small leads us to associate to our S-club system \bar C, a system of ideals \bar I=\langle I_\delta : \delta \in S\rangle as follows.

Each I_\delta is an ideal over C_\delta generated by the following sets:

  1. acc( C_\delta);
  2. \{\alpha\in C_\delta : \alpha <\beta\} for each \beta<\delta;
  3. \{\alpha\in C_\delta : cf(\alpha)<\gamma\} for each regular \gamma<\delta;

Then we see that the manner in which we want \bar C to guess clubs is that we need C_\delta\cap E\notin I_\delta stationarily-often for any club E\subseteq\lambda. So let’s say that the pair (\bar C, \bar I) guesses clubs if this happens. Then the statement of the theorem becomes:

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 the pair (\bar C, \bar I) guesses clubs.
  3. For each \delta\in S, the set of regular Jónsson cardinals below \delta is in I_\delta.

Then \lambda 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 M inconsideration sees enough points of nacc(C_\delta)\cap E where E was a particular club subset of \lambda. So we didn’t need the full force of condition 3. If we let A=\{ \alpha < \lambda : \alpha \text{ is regular and not J\'onsson}\}  we only needed to produce a club guessing sequence \bar C=\langle C_\delta : \delta \in S\rangle such that:

For every club E\subseteq \lambda, there are stationarily-many \delta\in S such that E\cap A\cap C_\delta\notin I_\delta.

In other words, we are asking that \langle C_\delta \cap A : \delta\in S\rangle 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 S\subseteq\lambda be stationary, \bar C=\langle C_\delta : \delta \in S\rangle be an S-club system, and let \bar I=\langle I_\delta : \delta \in S\rangle be a sequence of ideals (with I_\delta an ideal on C_\delta). We define the ideal id_p(\bar C, \bar I) by saying that A\in id_p(\bar C, \bar I) if and only if there is some club E\subseteq\lambda such that for every \delta\in S\cap E, we have E\cap A\cap C_\delta\in I_\delta.

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

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 the pair (\bar C, \bar I) guesses clubs.
  3. The set of regular cardinals which carry a Jónsson algebra is not in id_p(\bar C, \bar I).

Then \lambda 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.

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