Club Guessing in the Prikry Extension

I want to use this post to construct club guessing sequences in the extension by vanilla Prikry forcing over a measurable. This was definitely known before-hand (as Todd pointed out to me), and I think this sort of thing has gotten around mostly by word of mouth. So I figured I may as well write it down since the argument is so short.

So let \kappa be measurable, and let U be a witnessing normal measure on \kappa. Let S=(S^{\kappa^+}_\kappa)^V, and for each \delta\in S fix an increasing and continuous f_\delta:\kappa\to\delta with f_\kappa=id_\kappa. In particular, f_\delta puts \kappa in bijection with a club subset of \delta, and so U gets copied up to a normal measure U_\delta over that club. Now let \mathbb P be Prikry forcing over \kappa with respect to U, and let C be the generic Prikry sequence.

In V[C], let C_\delta=f_\delta ''C for each \delta\in S, and set \bar C=\langle C_\delta : \delta\in S\rangle. Recall that \mathbb P is \kappa^+-cc, and so every club subset of \kappa^+ in the extension contains a ground model club (so in particular S is stationary in V[C]). Next note that for every \delta\in S, the set C_\delta is a Prikry sequence for \delta, and in particular will diagonalize all club subsets of \delta in the ground model.

Claim: In V[C], for any club E\subseteq\kappa^+, there is a club D\subseteq \kappa^+ such that, for every \delta\in S\cap D, we have C_\delta\subseteq^*D. Here \subseteq^* is inclusion modulo bounded.

Proof: Let E\subseteq\kappa^+ be club, and let D\subseteq E be a ground model club contained in E. Let acc(D) be the set of accumulation points of D. If \delta\in S\cap acc(D), then D\cap\delta is a club subset of \delta which is in the ground model, and so C_\delta\subseteq^*D\cap\delta\subseteq E.

Okay now (still working in the extension), let’s consider the cofinality map cf\upharpoonright\kappa^+:\kappa^+\to\kappa. We may as well assume that every point of C is a regular cardinal. Since each f_\delta was continuous, note that cf''\bigcup_{\delta\in S}C_\delta=C is just our original Prikry sequence. So, we’re in the rather bizarre situation where the points that guess clubs all concentrate on a cofinal, progressive \omega-sequence of regular cardinals. On interesting thing to do is look at pcf(C), as that would give us some insight as to how scales and club guessing sequences might interact. The following is a theorem of Jech:

Thm: Let j:V\to M be a non-trivial elementary embedding where U came from. Then tcf(\prod C/J^{bd})=cf(j(\kappa)).

So here’s a question that’s probably much easier than I’m making it: Suppose I blow up 2^\kappa=\kappa^{++} before singularizing \kappa. Is there any way in which I can make sure that I can find j:V\to M with cf(j(k))=\kappa^{++}? If so, that’ll mean that I can guess clubs in such a way that the image of the ladder system under the cofinality map does not carry a scale. That would be kind of interesting.


Jonsson cardinals and better scales

I want to use this space to sketch the proof of the following known result:

Proposition: If \mu^+ is Jónsson for \mu singular, then there is B\subseteq\mu\cap REG cofinal in \mu with |B|=cf(\mu) such that \prod B carries a better scale.

All one needs to do here is show that pp(\mu)>\mu^+ and then use that to construct a better scale. So, the Jónsson assumption is moderately silly.

Proof: Let \lambda=\mu^+. We begin by showing that pp(\mu)>\mu^+. This is pretty standard, so fix a scale (\vec f, A) with |A|=cf(\mu) and we will show that co-boundedly many elements of A must be Jónsson. Otherwise, thin out A so that every element carries a Jónsson cardinal. Let \theta be large enough regular and let M\prec (H(\theta),\in,<_\theta, \vec f, A) be such that \lambda\in M, |M\cap\lambda|=\lambda, cf(\mu)+1\subseteq A and \lambda\not\subseteq M.

It suffices to show that \sigma\subseteq M for each sufficiently large \sigma<\mu. So fix \sigma, and note that we can find a\in A such that \sup M\cap a=a. Otherwise, Ch_M(a)<a for all large enough a and so Ch_M^A\in\prod A/J^{bd}. But then, there is some \alpha<\lambda with \alpha\in M such that Ch_M^A<^* f_\alpha. But this is silly, as A\subseteq M and so there is some a\in A with Ch_M(a)<f_\alpha(a)<a, but f_\alpha(a)\in M as it’s definable from parameters. So now we can find a>\sigma with a\in M and \sup(M\cap a)=a, and such that a carries a Jónsson algebra. But then, a\subseteq M. As \sigma<\mu was arbitrary, we get a contradiction.

So now we let A\subseteq \mu be such that |A|=cf(\mu), and every a\in A carries a Jónsson cardinal. Then, \mu^+\notin pcf_{J^{bd}}(A), otherwise we could find B=B_{\mu^+} a generator for J_{<\mu^+}^{J^{bd}}. In particular, pcf_{J^{bd}}(B)=\{\mu^+\} (since \min pcf_{J^{bd}}(B)=\mu^+) and so tcf(\prod B/J^{bd})=\mu^+, which is absurd by our earlier argument. Thus, we have that \min pcf_{J^{bd}}(A)>\mu^+ and in particular we get that \prod B/J^{bd} is \mu^{++}-directed. With this in hand, the rest of the proof is pretty easy.

We begin by fixing a silly square sequence \bar C=\langle C_\alpha : \alpha<\mu^+\rangle. We inductively construct a sequence \langle f_\alpha : \alpha<\mu^+\rangle as follows:

Stage \alpha=0: Just let f_0 be any function in \prod B.

Stage \alpha+1: Here we let f_{\alpha+1}>f_\alpha be any pointwise larger function in \prod B.

Stage \alpha limit: Now we have to do work. For each c\in C_\alpha, let h_c(a)=\sup_{\beta\in c}f_\beta(a) if ot(c)< a, and 0 otherwise. Note that these functions are non-zero almost everywhere. Then we let f_\alpha\in\prod B be such that h_c<^*f_\alpha for each c\in C_\alpha (here is where we use the directedness as |C_\alpha|\leq\mu^+).

Note that \vec f=\langle f_\alpha : \alpha<\mu^+ has an exact upper bound by construction h with the property that \{a\in A : cf(h(a))<\theta\} is bounded for each \theta<\mu. First we claim that \vec f is better in \prod_{a\in A}h(a). In other words, we need that:

for every \gamma of the appropriate cofinality, there is a club C\subseteq \gamma such that for every \alpha_1 in C, there is a a_{\alpha_1} such that \alpha_0<\alpha_1 in C implies that f_{\alpha_0}(a)<f_{\alpha_1}(a) for every a>a^* in A.

This follows directly from the construction, so let \gamma<\mu^+ be of cofinality above cf(\mu). Then let c\in C_\gamma be club, and let \alpha_0<\alpha_1\in acc(C). Note then that c\cap \alpha_1\in C_{\alpha_1} so let ot(c)<a^*\in A be such that h_{c\cap\alpha_1}(a)<f_{\alpha_1}(a) for all a>a^*. In particular, we see that f_{\alpha_0}(a)<f_{\alpha_1}(a). So we have betterness.

Then the usual argument shows that (by thinning out A if necessary) \prod_{a\in A}cf(h(a)) carries a better scale, which is what we actually want.

Club Guessing At Inaccessibles

Disclaimer: So after having typed this up, I’m a little shaky on a few details here still (mainly with keeping track of indices). But, I think I now have a better idea of how this proof should look if I attempt to rewrite it at some point in the future.

In this post, I want to work through the construction of a club guessing sequence which appears as Claim 0.14 of Sh413. Recall:

Definition: Let \lambda be a regular uncountable cardinal, and 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.

For the construction, we need a few definitions which have become common-place in the literature:

Definition: If C is a club set of ordinals, and \alpha\in nacc(C), set

Gap(\alpha, C)=(\sup(C\cap\alpha), \alpha).

Definition: If C and E are sets of ordinals with E\cap \sup(C) closed in \sup(C), we define:

Drop(C,E)=\{\sup(\alpha\cap E) : \alpha\in C\setminus\min(E)+1\}

Roughly what we’re doing here is dropping ordinals from C above min(E) down to their sup in E in order to create a closed subset of \sup(C).

Definition: Let C, E\subseteq\lambda for a regular uncountable, and let \langle e_\alpha : \alpha<\lambda\rangle be a \lambda-club system. For each \alpha\in nacc(C)\cap acc(E), define

Fill(\alpha, C, E)=Drop(e_\alpha, E)\cap Gap(\alpha, C)

We suppress mention of the \lambda-club system in our notation for Fill because it doesn’t really matter. What’s important is that Fill(\alpha, C, E) gives us a way of looking at the gap between sup (C\cap\alpha) and \sup (E\cap\alpha)=\alpha, and filling it in with a club subset of Gap(\alpha, C). So we will just think of \langle e_\alpha : \alpha\in \lambda\rangle as a fixed collection of clubs which we will draw upon to fill in gaps in an appropriate way.

Now, these describe operations which Shelah refers to as $gl^1$ and $gl^0$ in Cardinal Arithmetic. We’re going be using something called $gl^2$ over the course of our construction, which is the result of filling in multiple gaps and gluing the result together.

Definition: Let C, E\subseteq\lambda for a regular uncountable, and let \langle e_\alpha : \alpha<\lambda\rangle be a \lambda-club system. Further, let A\subseteq\lambda be a stationary set of limit ordinals. We define gl_n(C,E,A) by induction on n<\omega.

  • gl^2_0(C,E,A)=Drop(C,E)
  • gl^2_{n+1}(C,E,A)=gl_n(C,E,A)\cup\bigcup_{\beta\in nacc(gl^2_n(C,E,A))\setminus A}\big(Fill(\beta,E,gl^2_n(C,E,A))\cup\{\sup(\alpha\cap E) : \sup(\alpha\cap E)\geq\sup (\alpha\cap e_\beta)\}\big)
  • gl^2(C,E,A)=\bigcup_{n<\omega}gl^2_n(C,E,A).

This looks horrendous, but the main idea is that at each stage we look at the gaps in the previous stage outside of A, and fill those gaps in. Then at the end we glue it all together (hence gl) and provided that C is club in some \delta\in acc(E), the end result is a club of the same \delta. With this definition in hand, we’re ready to state the theorem and carry out the construction.

Claim 0.14 of Sh413: Assume

  1. \lambda is inaccessible.
  2. A\subseteq \lambda is a stationary set of limit ordinals such that, if \delta<\lambda and A\cap\delta is stationary in \delta, then \delta\in A.
  3. J is a \sigma-indecomposable ideal on \lambda extending the non-stationary ideal.
  4. S\notin J and S\cap A=\emptyset.
  5. \omega<\sigma=cf(\sigma)<\lambda and \delta\in S\implies cf(\delta)\neq\sigma.

Then for some S-club system \bar C=\langle C_\delta : \delta\in S\rangle we have that, for any E\subseteq\lambda:

\{\delta\in S : \delta=\sup(E\cap nacc(C_\delta)\cap A)\}\notin J.

Proof: We begin by fixing a \lambda-club system \bar e=\langle e_\alpha : \alpha<\lambda\rangle with the property that ot(e_\alpha)=cf(\alpha) and such that e_\delta\cap A=\emptyset for any limit \delta\in \lambda \setminus A (recall that A does not reflect outside of itself).

For any club E\subseteq\lambda, define C_{\delta,E}:=gl^2(e_\delta,E,A) if \delta\in acc(E), and C_{\delta,E}=e_\delta otherwise. If there is some club E of \lambda such that \bar C_E=\langle C_{\delta, E}: \delta\in S\rangle is as required, then we are done. So suppose otherwise. Then for very club E\subseteq\lambda, there is a club D(E)\subseteq\lambda such that

Y_E:=\{\delta\in S : \sup(D(E)\cap A\cap nacc (C_{\delta, E}))=\delta\in J.

Note that shrinking D(E) only makes Y_E smaller, so we may replace D(E) with D(E)\cap E in order to assume that D(E)\subseteq E. We now inductively pick sets E_\epsilon by induction on \epsilon<\sigma such that:

  1. E_\epsilon\subseteq\lambda is club;
  2. \xi<\epsilon\implies E_\epsilon\subseteq\ E_\xi;
  3. If \epsilon=\xi+1, then E_\epsilon\subseteq D(E_\epsilon).

This is simple. We first let E_0=\lambda, and at successor stages we take E_\epsilon=D_\xi. At limit stages we let E_\epsilon=\bigcap_{\xi<\epsilon}E_\xi.

Next, let E=\bigcap_{\epsilon<\sigma}E_\epsilon which is itself club in \lambda, and so E\cap A is stationary in \lambda. Thus, the set E'=\{\delta\in E : \delta=\sup(E\cap A\cap \delta)\} is club in \lambda. The following claim will finish the proof.

Claim: For every \delta\in S\cap E', there is some \epsilon_\delta<\sigma such that for every \epsilon_\delta\leq \epsilon<\sigma, we have \delta\in Y_{E_\epsilon}.

To see that this claim suffices, we begin by letting Y_{\epsilon}=\bigcap_{\epsilon\leq\xi<\sigma}Y_{E_\xi} for each \epsilon<\sigma. Of course, Y_\epsilon\in J as Y_\epsilon\subseteq Y_{E_\epsilon}\in J by assumption, and \epsilon_1<\epsilon_2\implies Y_{\epsilon_1}\subseteq Y_{\epsilon_2}. As J is \sigma-indecomposable, it’s closed under increasing unions of length \sigma and in particular \bigcup_{\epsilon<\sigma} Y_{\epsilon}\in J. On the other hand, the claim tells us that \delta\in Y_{\epsilon_\delta} for every \delta\in S\cap E', and so S\cap E'\in J. But by assumption we have that S\notin J, and as J extends the non-stationary ideal we have a contradiction.

Proof of Claim: We begin by fixing, for each \delta\in S\cap E', an increasing cofinal sequence \langle \beta^i_\delta : i<cf(\delta)\rangle of elements of A\cap E\cap S. Fix \delta\in S\cap E', and note that by assumption \delta\notin A, and so e_\delta\cap A=\emptyset. In particular, \{\beta^i_\delta : i<cf(\delta)\}\cap e_\delta=\emptyset. Now if we look at the construction of gl^2(e_\delta, E_\epsilon, A), we see that since \beta^i_\delta\in A\setminus e_\delta, we only add boundedly-many elements of \beta^i_\delta to gl^2_n(e_\delta, E_\epsilon, A) for each \epsilon<\sigma and i< cf(\delta).

For each i<cf(\delta), \epsilon<\sigma, and n<\omega, let \beta^i_\delta(n,\epsilon)=\min(gl^2_n(e_\delta, E_\epsilon, A)\setminus\beta^i_\delta) and note that since gl^2_n\subseteq gl^2_{n+1}, it follows that the sequence \langle \beta^i_\delta(n,\epsilon) : n<\omega\rangle is decreasing and hence eventually constant (say it stabilizes at n(i,\delta,\epsilon). Now, since \beta^i_\delta\cap gl^2_n(c_\delta, E_\epsilon, A) is bounded in \beta^i_\delta, it follows that \beta^i_\delta(n,\epsilon)\in nacc(C_{\delta, E_\epsilon}) for each n>n(\i,\delta,\epsilon). In fact, we also have that \beta^i_\delta(n,\epsilon) is in A for each n\geq n(i, \delta, \epsilon). Otherwise, we would have that \beta^i_\delta(n,\epsilon)\in nacc (gl^2_n(e_\delta, E_\epsilon, A)\setminus A, and so we would have ended up adding a club subset of \beta^i_\delta(n,\epsilon) in the next stage. So far we have shown: For each i<cf(\delta), and \epsilon<\sigma, there is some n(i,\delta,\epsilon)<\omega such that, for every \beta^i_\delta(n,\epsilon) we have

  1. \beta^i_\delta(\epsilon, n(i,\delta,\epsilon))=\beta^i_\delta(\epsilon,n) and
  2. \beta^i_\delta(\epsilon, n)\in nacc(C_{\delta, E_\epsilon})\cap A.

Next, we look at this sequence from the other direction. That is, we consider \langle \beta^i_\delta(\epsilon, n) : \epsilon<\delta\rangle for a fixed n. The best way to visualize this situation is that for each \delta\in S\cap E' and i<cf(\delta), we have a \sigma\times\omega matrix M_{i,\delta} where the \epsilon, n entry is \beta^i_\delta(\epsilon, n). So far, we’ve shown that the values are decreasing and eventually constant along the rows, and at this point we’d like to show the same for the columns. We begin by noting that the sequence of the sets E_\epsilonis decreasing, and so we have that \sup(\alpha\cap E_{\epsilon_1})<\sup (\alpha\cap E_{\epsilon_2}) whenever \epsilon_2>\epsilon_1. Therefore, the sequence \langle \beta^i_\delta(\epsilon, n) : \epsilon<\delta\rangle is decreasing and eventually stabilizes at some \epsilon(i,\delta, n)<\sigma. In particular, this tells us that

(\forall \xi>\epsilon(i,\delta, n))(\min (C_{\delta, E_\xi}\setminus \beta^i_\delta)=\min (C_{\delta, E_{\epsilon(i,\delta, n)}}\setminus \beta^i_\delta)=\beta^i_\delta(\epsilon(i,\delta, n), n))

Now, since \delta\in S, we know that cf(\delta)\neq\sigma and so for each \delta, we can find some \epsilon_\delta such that \sup\{i: \epsilon(i,\delta, n)\leq\epsilon_\delta \}=cf(\delta). This is pretty simple, as we just let \epsilon_\delta be the supremum of \epsilon(i,\delta, n) when running over i< cf(\delta). Then this supremum will be below \sigma as the cofinality of \delta is wrong.

At this point, we would like to show that for every \delta\in S\cap E', and \epsilon\geq\epsilon_\delta, we get that \delta\in Y_{E_\epsilon}. In particular, we need to show that \delta=\sup(D(E_\epsilon)\cap A\cap nacc(C_{\delta,E_\epsilon}). Recall that our construction of C_{\delta,E_\epsilon} actually gives us the elements of C_{\delta,E_\epsilon} are of the form \sup(\alpha\cap E_{\epsilon}. As D(E_\epsilon)\subseteq E_\epsilon, it follows that we only need to show that \delta=\sup(A\cap nacc(C_{\delta, E_\epsilon})). Recall that stabilization along columns tells us that if \epsilon_\delta\leq\epsilon<\sigma:

\min (C_{\delta, E_\epsilon}\setminus \beta^i_\delta)=\min (C_{\delta, E_{\epsilon(i,\delta, n)}}\setminus \beta^i_\delta)=\beta^i_\delta(\epsilon(i,\delta, n), n).

Finally, since \langle \beta^i_\delta : i<cf(\delta)\rangle is increasing and cofinal in \delta, and \beta^i_\delta<\beta^i_\delta(\epsilon,n)<\delta, it follows that:

\delta=\sup\{min(C_{\delta, E_\epsilon}\setminus \beta^i_\delta: \epsilon(i,\delta, n)\leq\epsilon_\delta\}.

Since we already showed that the supremum of the indices i involved in the above supremum has supremum equal to cf(\delta). Finally stabilization along the rows told us that eventually \beta^i_\delta(\epsilon, n) ends up in A\cap nacc C_{\delta, E_\epsilon} for all sufficiently large n<\omega. So, we can just pick the correct \beta^i_\delta(\epsilon, n) to achieve a supremum of \delta. That completes the proof of the claim, and consequently the proof of the whole thing.

Another Update

Okay, so I’ve actually finished working through the proof that any inaccessible Jónsson cardinal must be Mahlo (\omega-Mahlo even), but I haven’t had much time to write stuff up. The main issue is that I’ve also been working my way through Sh413, which contains the result that any \lambda which is an inaccessible Jónsson cardinal must be \lambda\times\omega-Mahlo.

What I’ll try to do is write up the proof that any inaccessible Jónsson cardinal must be Mahlo. That’s a bit more work than the thread I was following earlier, but not by much. One thing I will do is take the existence of the desired club guessing sequences for granted. I feel moderately comfortable in doing this because the construction of these sequences is incredibly similar to the construction in the case that \lambda=\mu^+ in EiSh819.

From there, I want to try and write up the result of working through the first section of Sh413. There’s a lot of material in that first section which ends up being tertiary to the main result, and I would like to write up a “straight shot” proof of the result. In particular, there is a really cool construction of a club guessing sequence provided that we have a stationary set S which does not reflect outside of itself (Claim 0.14 from Sh413). Actually I might write that part up sooner rather than later.

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.

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.

Scales from I[\lambda]

I want to make good on the promise of linking up scales to square-like principles that I made a couple of posts ago. In particular I want to sketch how one produces scales using the machinery of I[\lambda]. If we believe that I[\lambda] contains a lot of stationary sets, then most of the work was actually done by working through Claim 2.6A of Chapter 1 from Cardinal Arithmetic:

Theorem: Let I be an ideal on a set A of regular cardinals with \kappa>|A|^+ regular. Assume that:

  1. \lambda>\kappa^{++} is regular such that there is some stationary S\subseteq S^\lambda_{\kappa^+} which has a continuity condition \bar C;
  2. \vec f=\langle f_\alpha : \alpha<\lambda\rangle is a sequence of functions from A to the ordinals;
  3. \vec f obeys \bar C.

Then \vec f has a \leq_I-exact upper bound.

We just have to show that it’s we can construct sequences which obey continuity conditions, and then there’s a relatively standard argument which allows us to move immediately from an exact upper bound for an appropriate sequence to a scale. Let’s briefly recall what continuity conditions are, and how they relate to I[\lambda]:

Theorem (Shelah): Let \lambda be a regular cardinal. Then for S\subseteq \lambda, we have S\in I[\lambda] if and only if there is a sequence \bar C= \langle C_\alpha : \alpha<\lambda\rangle and a club E\subseteq\lambda such that:

  1. Each C_\alpha is a closed (but not necessarily unbounded) subset of \alpha;
  2. if \beta\in nacc(C_\alpha) then C_\beta=\alpha\cap C_\alpha;
  3. If \delta\in S\cap E, then \delta is singular, and C_\delta is a club subset of \delta of order type cf(\delta).

The sequence \bar C is called a continuity condition for S, and functions somewhat like a square sequence over S. The major difference is that we only have coherency on the non-accumulation points which is a significant weakening, but still allows them to be useful enough. So the fact that I[\lambda] contains a stationary subset of S^\lambda_\kappa for every regular\kappa such that \kappa^{++}<\lambda can be regarded as a weak fragment of square which is true in ZFC.

Since we want to produce scales, our focus will be on I[\mu^+] for \mu singular. In particular, we have that for every regular \kappa<\mu, there is a stationary S\subseteq S^{\mu^+}_{\kappa} such that S\in I[\mu^+]. Further, we also have that if \mu is strong limit, then S^{\mu^+}_{\leq cf(\mu)}\in I[\mu^+], though this won’t be particularly important to us (a proof of this can be found in Todd Eisworth’s Handbook chapter for the interested).

First, we show how to produce sequences that obey continuity conditions. So, fix a set of regular cardinals A\subseteq \mu cofinal in \mu with ot(A)=cf(\mu) and such that |A|^+<\min A. Following standard notation, we will let J^{bd}[A] denote the ideal of bounded subsets of A.

We first show that \prod A/J^{bd}[A] is \mu^+ directed. To see this, note that is suffices to show that \prod A/J^{bd}[A] is \mu-directed. For any set F\subseteq \prod A/J^{bd}[A] such that |F|=\mu, then we can rewrite F=\bigcup_{i<cf(\mu)}F_i where |F_i|<\mu. From there, we use \mu-directedness to produce bounds f_i\in\prod A/J^{bd}[A] for each F_i, and then bound \{f_i : i<cf(\mu)\} by f\in \prod A/J^{bd}[A]. For \mu-directedness, let F\subseteq \prod A/J^{bd}[A] be such that |F|<\mu. Then let f be defined by f(a)=\sup \{g(a) : g\in F\}, and note that f is defined almost everywhere since each a\in A is regular and A is cofinal in \mu. Clearly then f+1 is a <_{J^{bd}[A]}-upper bound for F.

Now let \kappa<\mu be regular, and let S\subseteq S^{\mu^+}_\kappa be such that S\in I[\lambda] with \bar C=\langle C_\alpha : \alpha<\lambda\rangle a witnessing continuity condition. We first recall what it means for a sequence \vec f=\langle f_\alpha : \alpha<\mu^+\rangle to obey \bar C:

Definition: We say \vec f weakly obeys \bar C if:

If \alpha<\lambda is such that ot(C_\alpha)\leq \kappa, then for each \beta\in nacc(C_\alpha), we have f_\beta(i)<f_\alpha(i) for each i<\kappa.

This definition looks like a weakening of the one originally given, but it’s all that was required for the proof of Claim 2.6A to go through. Now we inductively define a sequence \vec f which obeys \bar C as follows. We first let f_0 be any function in \prod A/J^{bd}[A]. At stage \alpha, we suppose that f_\beta has been defined for each \beta<\alpha. We let f_\alpha' be a <_{J^{bd}[A]}-upper bound for \{f_\beta : \beta<\alpha\} as guaranteed by \mu^+-directedness. If C_\alpha is empty or ot(C_\alpha)>\kappa, then we just set f_\alpha=f_\alpha'. Otherwise, we let f_\alpha be defined by setting f_\alpha(a)=\max\{f_\alpha'(a), \sup_{\beta\in C_\alpha}f_\beta(a)\}+1. Note that since \kappa<\mu, we know that f_\alpha(a) is defined almost everywhere. It is also clear by construction that \langle f_\alpha : \alpha<\mu^+\rangle is a <_{J^{bd}[A]}-increasing sequence which weakly obeys \bar C and so we are done.

I also want to note that we could have started with a fixed sequence \vec g=\langle g_\alpha : \alpha<\mu^+\rangle, and asked that not only \vec f weakly obey \bar C, but also that g_\alpha<f_{\alpha+1} for each \alpha<\mu^+. So for example if \vec g weakly obeyed some other continuity condition \bar D, then the resulting \vec f would weakly obey both \bar D and \bar C. Further, if \vec f obeys a continuity condition for a stationary subset of S^{\mu^+}_\kappa, then one can show that the exact upper bound f produced by Claim 2.6A satisfies:

\{\ a\in A : cf(f(a))<\kappa\}\in J^{bd}[A].

 Okay, with all of this in hand, we can produce a <_{J^{bd}[A]}-increasing sequence of functions \vec f=\langle f_\alpha : \alpha<\mu^+\rangle in \prod A/J^{bd}[A] with the following properties:

  1. \vec f has an exact upper bound f;
  2. For every regular \kappa with \min(A)\leq \kappa<\mu, the set \{a\in A : cf(f(a))<\kappa\}\in J^{bd}[A].

So, we then have that sequence \vec f witnesses that \prod_{a\in A} f(a)/J^{bd}[A] has true cofinality \mu^+. Now, by possibly altering f on a null set we may assume \min \{f(a):a\in A\}>|A|. Let B=\{cf(f(a)) : a\in A\}, and note by condition 2, that B is cofinal in \mu and has order type cf(\mu). A relatively standard argument then allows us to conclude that tcf (\prod B/J^{bd}[B])=\mu^+, and letting the witnessing sequence be \vec h=\langle h_\alpha : \alpha<\lambda\rangle, we get that (B,\vec h) is a scale on \mu.

Honestly, parts of this sketch are pretty bare-bones, but the idea was to show that Claim 2.6A (once appropriately modified) is the only really difficult part behind producing scales. In fact, that claim plays the same role that the trichotomy theorem does for the theory of exact upper bounds. In particular, it shows us that, provided we can construct certain sorts of sequences, we can then get nice exact upper bounds. It just turns out that these sequences, once we have enough of the I[\lambda] combinatorics in hand, are relatively easy to produce. From there, it’s just standard arguments showing that we really only need exact upper bounds to do a lot of the things we want. An alternative approach to exact upper bounds (outside of I[\lambda] or trichotomy) is also furnished through what Abraham and Magidor call (*)_\kappa. It turns out that (*)_\kappa is incredibly similar to having continuity conditions for a stationary subset of S^\lambda_\kappa lying around.

Overall though, all three of these approaches are doing roughly the same thing.