Universal Cofinal Sequences and The Weak pcf Theorem (Sh506)

This post follows up the previous one.

As before, we will let \mu be a singular cardinal, A=\langle \mu_i : i<\mu\rangle a sequence of regular cardinals with limit \mu, and I be a fixed ideal over \mu such that wsat(I)<min(A). Under these assumptions, we’ve shown that, letting

J_{<\lambda}^I[A]=\{B\subseteq A : for every D ultrafilter over \mu disjoint from I, if B\in D, then cf(\prod A/D)<\lambda \}\cup I

If \lambda\geq wsat(I), and J_{<\lambda} is proper, then \prod A/J_{<\lambda} is \lambda-directed. This was lemma 1.5 in Sh506. Our goal now is to use this to prove that universal cofinal sequences exist, which gives us (modulo a claim from Sh506) what Shelah calls the weak pcf theorem. Let’s take a little bit to derive some corollaries from \lambda-directedness (these appear in Sh506 as well). First note that the lemma holds trivially when \lambda<wsat(I) since wsat(I)<min(A), as well as when \mu\in J_{<\lambda} since then \prod A/J_{<\lambda} has one lement. Recall that

pcf_I(A)=\{\lambda=cf(\prod A/D) : D is an ultrafilter over \mu disjoint from I\}. (note that every \lambda\in pcf_I(A) is regular)

Corollary: Under our current assumptions, for every ultrafilter D over \mu, cf(\prod A/D)\geq\lambda if and only if J_{<\lambda}^I[A]\cap D=\emptyset.

Proof: By definition, cf(\prod A/D)\geq\lambda implies that J_{<\lambda}^I[A]\cap D=\emptyset so this direction is uninteresting. For the other direction, suppose that J_{<\lambda}^I[A]\cap D=\emptyset, then \prod A/D is \lambda directed and so cf(\prod A/D)\geq \lambda.

This corollary tells us that cf(\prod A/D)=\lambda if and only if D has non-empty intersection with J_{<\lambda^+}, but misses J_{<\lambda}. So cf(\prod A/D)=\lambda if and only if \lambda is the first cardinal for which D has nontrivial intersection with J_{<\lambda^+}. It follows that we can associate to each \lambda\in pcf_I(A) a set X_\lambda\in J_{<\lambda^+}\setminus J_{<\lambda}, which gives us an injection from pcf_I(A) to \mathcal{P}(\mu). The idea then is that, if we can get some control over how we generate J_{<\lambda^+} from J_{<\lambda}, we should be able to say something more about the size of pcf_I(A). Also, note that each of these sets X_\lambda is I-positive.

We also have that max pcf_I(A) exists. To see this, let

J=\bigcup\{J_{<\lambda} : \lambda\in pcf_I(A)\}, and note that J is a proper ideal since it is the union of an ascending chain of proper ideals (as \lambda\in pcf_I(A) implies that J_{<\lambda} is proper). So, let D be an ultrafilter disjoint from J and let \kappa=cf(\prod A/D). As D\cap J_{<\lambda}=\emptyset for all \lambda\in pcf_I(A), it follows that \kappa\in pcf_I(A) but \kappa\geq \lambda for all \lambda\in pcf_I(A). Hence, \kappa=maxpcf_I(A).

Going back to the pcf ideals, we need a definition from Sh506:

Definition: We say that \lambda\in pcf_I(A) is semi-normal if there are B_\alpha for \alpha<\lambda such that:

  1. If \alpha <\beta, then B_\alpha \subseteq_{J_{<\lambda}}B_\beta, and
  2. J_{<\lambda^+}=J_{<\lambda}+\{B_\alpha : \alpha<\lambda\}.

Here the statement in 2. simply means that J_{<\lambda^+} is generated from J_{<\lambda} and \{B_\alpha : \alpha<\lambda\}. We also need a definition from the Abraham-Magidor chapter in the handbook.

Definition: Let \vec{f}=\langle f_\xi : \xi<\lambda\rangle be a <_{J_{<\lambda}}-increasing sequence in \prod A. We say that \vec{f} is a universal cofinal sequence for \lambda. if, for any ultrafilter D over \mu, if cf(\prod A/D)+\lambda, then the sequence \vec{f} is cofinal in \prod A/D.

We will take the following for granted:

Fact 2.2(2) of Sh506: \lambda\in pcf_I(A) is semi-normal if and only if there is a universal cofinal sequence for \lambda.

The following is buried in Sh506 (probably in the proof of Lemma 2.6), but we will instead mimic the proof of lemma 1.5 from Sh506. The proof here is also very similar to the proof that universal cofinal sequences exist given in the Abraham-Magidor chapter (except that one is for the original case).

Theorem: Every \lambda\in pcf_I(A) has a universal cofinal sequence.

Before proceeding, we should note that in the proof of Lemma 1.5, we actually made use of the following fact (really we kind of proved it):

I is \theta-weakly saturated if and only if every \subseteq-increasing \theta-sequence \langle A_i : i<\theta\rangle of I-positive sets is eventually constant modulo I.

The proof of this will be very similar to the proof of Lemma 1.5, insofar as we will proceed by induction on \alpha<wsat(I), and suppose that we fail to get a universal cofinal sequence at each stage. From this we will be able to produce a contradiction to weak saturation. We begin by noting that, if \lambda=\min (A), then we can define \vec{f}=\langle f_\xi : \xi<\lambda\rangle by setting f_\xi(i)=\xi which is an everywhere increasing sequence. So, it trivially is a universal sequence. In that vein, we may assume that wsat(I)<min(A)<\lambda.

We will proceed by induction on \alpha<wsat(I), and construct candidate universal sequences \vec{f}^\alpha=\langle f_\xi^\alpha : \xi<\lambda. Now, we want to (as in the proof of Lemma 1.5 from Sh506) come up with sets B^\alpha_\xi that are \subseteq-increasing in the \alpha coordinate but differ from each other modulo J_{<\lambda} (and hence I). So we will ask that not only is the collection \langle f^\alpha_\xi : \alpha<wsat(I), \xi<\lambda\rangle strictly increasing modulo J_{<\lambda} in the xi coordinate, but that it is \leq-increasing in the \alpha coordinate. With that in mind, we will use \lambda-directedness to inductively construct these sequences.

For \alpha=0, we let \vec{f}^0=\langle f^0_\xi : \xi<\lambda\rangle be any <_{J_{<\lambda}}-increasing sequence in \prod A. We can create such a sequence inductively as follows: let f^0_0 be arbitrary, and then assume that f^0_\eta have been defined for \eta<\xi. By \lambda-directedness, we can find g\in\prod A such that f_\eta^0\leq_{J_{<\lambda}} g for all \eta<\xi, and let f^0_\xi=g+1.

At limit stages, let \gamma<\lambda and assume that \vec{f}^\alpha has been defined for each \alpha<\gamma. We inductively define \vec{f}^\gamma=\langle f^\gamma_\xi : \xi<\lambda as follows: let f^\gamma_0=\sup \{f^\alpha _0 : \alpha<\gamma\}, which is in \prod A since \gamma<wsat(I)<min(A). Now suppose that f^\gamma_\eta has been defined for each \eta<\xi, and let g=\sup \{f^\alpha_\eta : \alpha<\gamma\}. Again g\in \prod A, and let h be such that f^\gamma_\eta\leq_{J_{<\lambda}} h for all \eta<\xi by \lambda-directedness. Then define f^\gamma_\xi by f^\gamma_\xi(i)=max\{g(i),h(i)\}+1, which is as desired.

At successor stages suppose that \vec{f}^\alpha has been defined. If \vec{f} is a universal sequence, then we can terminate the induction. If not, we inductively define \vec{f}^{\alpha+1}=\langle f^{\alpha+1}_\xi: \xi<\lambda\rangle as follows: Since \vec{f}^\alpha is not universal, we can find an ultrafilter D_\alpha over \mu with the property that cf(\prod A/D_\alpha)=\lambda, but \vec{f}^\alpha is <_{D_\alpha}-dominated by some h\in\prod A/D_\alpha (note that D_\alpha is disjoint from J_{<\lambda}). Let \vec{g}=\langle g_\xi : \xi<\lambda\rangle be a cofinal sequence in \prod A/D_\alpha. We define f^{\alpha+1}_0 by setting f^{\alpha+1}_0(i)=max\{h(i), f^{\alpha}_0(i), g_0(i)\}. Now suppose that f^{\alpha+1}_\eta has been defined for each \eta<\xi, and let h be such that f^{\alpha+1}_\eta\leq_{J_{<\lambda}} h for all \eta<\xi by \lambda-directedness. Then define f^{\alpha+1}_\xi by f^{\alpha+1}_\xi(i)=max\{f^\alpha_\xi(i),h(i), g_\xi(i)\}+1, which is as desired. Note that \vec{f}^{\alpha+1} is cofinal in \prod A/D_\alpha

We claim that we must have terminated the induction at some stage. Otherwise, we will have defined for each \alpha<wsat(I) the following:

  1. Sequences \vec{f}^\alpha=\langle f^\alpha_\xi : \xi<\lambda\rangle which are J_{<\lambda}-\increasing in the \xi coordinate, and \leq-increasing in the \alpha coordinate.
  2. Ultrafilters D_\alpha disjoing from J_{<\lambda} such that \vec{f}^alpha is <_{D_\alpha} dominated by f^{\alpha +1}_0, and \vec{f}^{\alpha+1} is cofinal in \prod A/D_\alpha.

We will use this to derive a contradiction. We begin by letting h\in\prod A be defined by setting h(i)=sup\{f_0^\alpha : \alpha<wsat(I)\} (recall that wsat(I)<min(A)). By condition 2 above, for every \alpha<wsat(I), there exists an index \xi(\alpha)<\lambda such that h <_{D_\alpha} f_{\xi(\alpha}^{\alpha +1}. Since wsat(I)<min(A)<\lambda for \lambda regular, it follows that \xi(*)=sup\{\xi(\alpha) : \alpha<wsat(I)\} is below \lambda. So, for each \alpha<wsat(I), we have that h<_{D_\alpha} f_{\xi(*)}^{\alpha+1}. Now define the sets

B_\alpha =\{i<\mu : h(i) \leq f^\alpha_{\xi(*)}(i)\}.

By construction, we have that B_\alpha\notin D_\alpha since f_{\xi(*)}^\alpha <_{D_\alpha} f_0^{\alpha+1}\leq h. On the other hand, B_{\alpha+1}\in D_\alpha since h<_{D_\alpha} f^{\alpha+1}_{\xi(*)}. So, it follows that B_\alpha\neq B_{\alpha+1} modulo J_{<\lambda} (hence modulo I). But since f^\alpha_{\xi(*)}\leq f^{\alpha+1}_{\xi(*)}, we have that B_\alpha\subseteq B_{\alpha+1} (in fact \beta<\alpha implies that B_\beta\subseteq B_\alpha) and so we are in the same position as the proof of Lemma 1.5. That is, \langle B_\alpha\setminus B_{\alpha+1} : \alpha<wsat(I)\rangle is a collection of I-positive sets which are disjoint, contradiction weak saturation. Therefore, the induction must have halted at some stage and we are done.

Therefore, we have the following result:

Under our current assumptions, each \lambda\in pcf_I(A) is semi-normal. So, we have what Shelah refers to as the weak-pcf theorem. Our next goal is to show that this can be improved to full normality (i.e. the pcf theorem) under stronger hypotheses for our ideal I.

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