Notes on Sh506 (part I)

In pcf theory, we look at reduced products \prod A/I where A is a set of regular cardinals with singular limit \mu satisfying min(A)>|A| (such sets are referred to as progressive in the literature). Of course, if \mu is an \aleph-fixed point, we cannot ask that A is a progressive interval of regular cardinals, which is also a required hypothesis of many theorems of pcf theory. Part of the point of Sh 506 is to allow us to do pcf theory at \aleph-fixed points, which interests me because this may be applicable to the problem of finding a Jónsson successor of a singular. So, I’m going to work through that paper using this blog. We first need a few definitions.

If I is an ideal over a set A and \theta is a cardinal, then we say that I is \theta-weakly saturated if X cannot be partitioned into \theta-many I positive sets. We say that wsat(I) is the smallest cardinal \theta for which I is \theta-weakly saturated. Throughout, \mu will denote a fixed singular cardinal with A=\langle \mu_i : i<\mu\rangle a sequence of cardinals with sup A=\mu, and I an ideal over \mu.

Definition:
pcf_I(A)=\{\lambda=cf(\prod A/D): D is an ultrafilter disjoint from I\}.

J_{<\lambda}^I[A]=\{B\subseteq \mu : if D is an ultrafilter over \mu extending I with B\in D, then cf(\prod A/D)<\lambda\}\cup I.

For the sake of simplicity, we will just use J_{<\lambda} to refer to J_{<\lambda}^I[A] when I and A are clear from the context. One of the main results of pcf theory is the existence of generators for these pcf ideals provided that A is a progressive set of regular cardinals, and instrumental in that is the fact that \prod A/J_{<\lambda} is \lambda-directed. It turns out that we can weaken the assumption that A is progressive.

Theorem(Lemma 1.5 of Sh 506): Assume that A is a sequence of regular cardinals with the property that min(A)>wsat(I), then if \lambda\geq wsat (I) is a cardinal with J_{<\lambda}^I[A] proper, then \prod A/J_{<\lambda} is \lambda-directed.

Proof: We will inductively show that, by induction on \lambda_0 <\lambda that \prod A/J_{<\lambda} is \lambda_0^+ directed. If |F|\leq wsat(I)<min(A), then we let g be defined by g(i)=sup\{f(i) : f\in F\}. Then since each \mu_i is regular, it follows that g\in \prod A and f\leq g everywhere.

By way of induction, assume we have shown, for wsat(I)<\lambda_0<\lambda a cardinal, that \prod A/J_{<\lambda} is \lambda_0-directed, and let F\subseteq \prod A of size \lambda_0 be given. We first assume that \lambda_0 is singular. In this case, we can write F=\bigcup_{\alpha<cf(\lambda_0)}F_\alpha such that |F_\alpha|<\lambda_0. Then by assumption, we can bound each F_\alpha by some g_\alpha, and then bound the set \{ g_\alpha : \alpha<cf(\lambda_0)\} by some g\in \prod A. We then have that f\leq g modulo J_{<\lambda} for each f\in F.

So assume that \lambda_0 is regular. We begin by replacing F=\{h_i : i<\lambda_0\} with a \leq_{J_{<\lambda}}-increasing sequence \vec{f}=\langle f_i : i<\lambda_0\rangle. We just let f_i be a \leq_{J_{<\lambda}}-upper bound for \{h_j : j\leq i\}\cup \{f_j : j<i\}. By construction, if we can find a g\in \prod A such that f_i \leq g modulo J_{<\lambda} for each i<\lambda_0, then we will be done. At this point, we will proceed by induction on \alpha <wsat(I) and attempt to construct a \leq_{J_{<\lambda}}-increasing sequence of candidates for bounds of \vec{f}. As usual, we will show that this construction must terminate at some point, or we will be able to generate a contradiction.

By induction on \alpha <wsat(I), we will define functions g_\alpha, ordinals \xi(\alpha), and sequences \langle B^\alpha_\xi : \xi <\lambda_0\rangle with the following properties:

  1. g_\alpha\in \prod A and for all \beta<\alpha, we have that g_\beta\leq g_\alpha;
  2. B_\xi^\alpha:=\{i<\mu : f_\xi(i)>g_\alpha (i)\};
  3. For each \alpha <wsat(I), and every \xi\in[\xi(\alpha+1),\lambda_0), we have that B^\alpha_\xi\neq B^{\alpha+1}_\xi modulo J_{<\lambda}.

The construction proceeds as follows. At stage \alpha=0, we simply let g_0=f_0, and set \xi(\alpha)=0 (note that \xi(\alpha) only matters when \alpha=\beta+1 for some ordinal \beta<wsat(I)). At limit stages, assume that g_\beta has been defined for each \beta<\alpha, and define g_\alpha by setting g_\alpha(i)=sup_{\beta<\alpha} f_\beta(i). Note since \alpha<wsat(I)<min(A) and each \mu_i is regular, that g_\alpha\in\prod A.

At successor stages, let \alpha=\beta+1, and suppose that g_\beta has been defined. If g_\beta is a \leq_{J_{<\lambda}}-upper bound for \vec{f}, then we’re done and we can terminate the induction. Otherwise, note that the sequence \langle B^\beta_\xi : \xi <\lambda_0\rangle is \subseteq_{J_{<\lambda}}-increasing and so there is a minimum \xi(\alpha) for which every \xi\in[\xi(\alpha),\lambda_0) has the property that B_\xi^\beta\notin J_{<\lambda} (else if there is no such \xi(\alpha), then g_\beta was indeed the desired bound). By definition, that means we can find some ultrafilter D, disjoint from J_{<\lambda} such that B_{\xi(\alpha)}^\beta\in D and cf(\prod A/D)\geq\lambda. Thus is follows that \vec{f} must have a <_D-upper bound in \prod A, say h_\alpha. We then define g_\alpha\in \prod A by g_\alpha (i)=max\{g_\beta(i), h_\alpha(i)\}.

Note that for each \xi\in[\xi(\beta+1),\lambda_0), we have that B_\xi^\beta\in D. On the other hand, our definition of g_\alpha gives us that B_\xi^{\beta+1}\notin D since g_\alpha is at least h_\alpha everywhere. Thus, condition 3 is satisfied, as are 1 and 2 trivially by construction.

We claim that this process must have terminated at some stage. Otherwise, we let \xi(*)=\sup\{\xi(\alpha) : \alpha<wsat(I)\}, and note that each B^\alpha_{\xi(*)}\notin J_{<\lambda} since the induction never terminated. Next, we note that condition 1) gives us that for \alpha\leq \beta, we have B_{\xi(*)}^\beta\subseteq B_{\xi(*)}^\alpha and so B_{\xi(*)}^\alpha\setminus B_{\xi(*)}^{\alpha+1}\notin J_{<\lambda}. Therefore, for \alpha<\beta, we have that B_{\xi(*)}^\beta\subseteq B_{\xi(*)}^{\alpha+1} and so the sets B_{\xi(*)}^\alpha\setminus B_{\xi(*)}^{\alpha+1} and B_{\xi(*)}^\beta\setminus B_{\xi(*)}^{\beta+1} are disjoint I-positive sets (since J_{<\lambda} extends I). But then we have a partition \{B_{\xi(*)}^\alpha\setminus B_{\xi(*)}^{\alpha+1} : \alpha < wsat(I)\} of \mu into wsat(I)-many disjoint I-positive sets, which is a contradiction. Therefore the process terminated at some point and \vec{f} (hence F) has a \leq_{J_{<\lambda}}-upper bound. This completes the induction and the proof.

 

A couple of observations:

Since \lambda-directedness seems to be the key ingredient in obtaining universal cofinal sequences, I suspect that we can recover such sequences for each \lambda in pcf_I(A). Further, the proof has a lot of the flavor of Shelah’s trichotomy theorem, especially the bit where we used witnessing ultrafilters to generate upper bounds for \vec{f}. So, I again suspect that one can prove a variant of the Trichotomy theorem for these pseudo-progressive sets of regular cardinals when we have a sufficiently nice ideal laying about. I unfortunately haven’t had time to work through the details and write this stuff down, but hopefully I’ll get around to it in the near future.

Also, it’s relatively easy to see that we get the usual characterization that for an ultrafilter D over A disjoint from I, we have cf(\prod A/D)\geq \lambda if and only if J_{<\lambda}^I[A]\cap D=\emptyset. From there, we also get a natural correspondence between \lambda\in pcf_I(A) and the existence of a set X_\lambda\in J_{<\lambda^+}\setminus J_{<\lambda}. We can also use this to show that pcf_I(A) has a maximum element. So, we recover a lot of nice information about the structure of pcf_I(A) from just this lemma. On the other hand, we don’t necessarily get that pcf_I(A) must be an interval of regular cardinals since that requires what Abraham and Magidor call the “no holes argument”. The only proof of that theorem that I know heavily utilizes the theory of exact upper bounds, which would need the trichotomy theorem (or something equivalent to it).

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