# 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). 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 since $wsat(I), 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, 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).

We will proceed by induction on $\alpha, 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 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. 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 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 (recall that $wsat(I)). By condition 2 above, for every $\alpha, there exists an index $\xi(\alpha)<\lambda$ such that $h <_{D_\alpha} f_{\xi(\alpha}^{\alpha +1}$. Since $wsat(I) for $\lambda$ regular, it follows that $\xi(*)=sup\{\xi(\alpha) : \alpha is below $\lambda$. So, for each $\alpha, 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 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 $I$.