# A short post on Generators

So one of my goals for this semester (or year) is to try and figure out what’s going on in Section 2 of Sh460. Of course, the section starts off by referencing Claim 6.7A of Sh430 and improving it (without mentioning what’s actually going on in that claim). Looking back at Claim 6.7A of Sh430, it turns out that this references some of the tools used in the proof of Claim 6.7, which gives us the existence of closed and transitive generators. Now it turns out that one of the things that we worked through in the summer school at UC Irvine (which I like to call pcf-fest 2016) is this very thing.

The proof that James gave was a bit different, but I think that claim 6.7A is really just making more explicit the relationship between transitive generators, universal sequences, and $\kappa$-IA elementary substructures. So what I’d like to do first is go back and work through the existence of transitive generators, and see how much of this stuff I can tease out along the way. Hopefully that’ll also put me in a good mindset to work through the Sh460 stuff. I figured that a good place to start is with the usual construction of generators and how they relate to universal sequences.

Throughout this, I’m going to let $A$ be a collection of regular cardinals, and put restrictions on it as necessary.

Definition: Let $A$ be a set of regular cardinals, and define

$pcf(A)=\{cf(\prod A, <_U): U\text{ is an ultrafilter on }A\}$

Here $<_U$ is just domination modulo $U$. I will frequently bounce between $(\prod A, <_U)$ and $\prod A/U$.

Definition: Let $\lambda$ be a regular cardinal, then

$J_{<\lambda}[A]=\{X\subseteq A : pcf(X)\subseteq\lambda\}$

Note that this is an ideal on $A$.

Definition: We say that $B_\lambda$ is a generator of $J_{<\lambda}$ if $\lambda\in pcf(A)$, and $J_{<\lambda^+}[A]$ is generated from $J_{<\lambda}[A]$ by $B_\lambda$.

In particular, we see that $J_{<\lambda^+}[A]=\{X\subseteq A : X\subseteq_{J_{<\lambda}[A]}B_\lambda\}$. Also note that if $\lambda\notin pcf(A)$, then obviously $pcf(X)\subseteq\lambda^+\implies pcf(X)\subseteq \lambda$ for $X\subseteq A$. So in the case that $\lambda\notin pcf(A)$, asking for a generator is fantastically uninteresting. Now, let’s say that $A$ is progressive whenever $|A|^+<\min(A)$.

Definition: Let $\lambda\in pcf(A)$, then $\vec f^\lambda=\langle f^\lambda_\alpha : \alpha<\lambda\rangle$ is a universal sequence for $\lambda$ if:

1. $\vec f^\lambda$ is $<_{J_{<\lambda}}$-increasing;
2. For any ultrafilter $U$ over $A$ such that $cf(\prod A/U)=\lambda$, we have that $\vec f^\lambda$ is cofinal in $\prod A/U$.

Note that if $U$ is an ultrafilter with $cf(\prod A/U)=\lambda$, then $U\cap J_{<\lambda}=\emptyset$. Otherwise, there is some $X\subseteq A$ with $pcf(X)\subseteq\lambda$ and $X\in U$. But then, $cf(\prod X/U)=cf(\prod A/U)$ since $X=_U A$, which would mean that $\lambda\in pcf(X)$. This gives us another characterization of $J_{<\lambda}[A]$ as the collection of subsets of $X$ which forces $cf(\prod A/U)<\lambda$ whenever they get assigned measure one by $U$.

Theorem (Shelah): If $A$ is progressive, then for every $\lambda\in pcf(A)$, there is a universal sequence for $\lambda$ with a $J_{<\lambda}$ exact upper bound $h\in{}^A ON$.

Why are universal sequences useful? Well, if $f^\lambda$ is a universal sequence for $\lambda \in pcf(A)$ with exact upper bound $h$, then the set $B_\lambda=\{a\in A : h(a)=a\}$ is actually a generator for $\lambda$. Now, these generators are only unique modulo $J_{<\lambda}$, and so we have some room to massage them. In the next post, I want to examine the possibility of doing just that.

# 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 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 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) 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 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.

# Continuity Conditions and I[\lambda]

In the previous post, I worked through a result of Shelah’s that allows us to produce exact upper bounds from continuity conditions. I want to use this post to briefly talk about where these things are coming from. As usual, for regular $\kappa<\lambda$, we denote:

$S^\lambda_\kappa=\{\alpha<\lambda : cf(\alpha)=\kappa\}$

Further, for a set $C$ of ordinals, we denote:

$acc(C)=\{\alpha\in C : \alpha=\sup (\alpha\cap C)\}$

$nacc(C)=C\setminus acc(C)$.

Definition: Let $\lambda$ be a regular cardinal, and let $\vec a=\langle a_\alpha : \alpha<\lambda\rangle$ be a sequence of elements of $[\lambda]^{<\lambda}$. Given a limit ordinal $\delta<\lambda$, we say that $\delta$ is approachable with respect to $\vec a$ if there is an unbounded $A\subseteq \delta$ of order type $cf(\delta)$ such that every initial segment of $A$ is enumerated prior to stage $\delta$. More precisely:

For every $\alpha<\delta$, there exists a $\beta<\delta$ such that $A\cap\alpha=a_\beta$.

Definition: Let $\lambda$ be a regular cardinal and define $I[\lambda]$ to be the collection of $S\subseteq\lambda$ such that there is a sequence $\vec a=\langle a_\alpha :\alpha<\lambda$ of elements of $[\lambda]^{<\lambda}$ and a club $E\subseteq\lambda$ such that every $\delta\in E\cap S$ is singular and approachable with respect to $\vec a$.

So the idea is that an ordinal is approachable with respect to some sequence above if there is some unbounded set whose initial segments get captured in a timely manner. A set of ordinals $S$ is in $I[\lambda]$ if almost every (modulo clubs) ordinal in $S$ is uniformly approachable, and this uniformity is captured by a single sequence.

Proposition: $I[\lambda]$ is a (possible improper) normal ideal over $\lambda$.

One thing to note is that, if $\lambda\in I[\lambda]$, then there is a club of singular ordinals which are all approachable by way of a single sequence $\vec a$. So one can imagine that if this is indeed possible, then $\lambda$ must have some nice combinatorial structure. It turns out that this is indeed possible, and this yields a square-like principle. The following alternative characterization of $I[\lambda]$ makes this more evident.

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 $cf(\delta)$.

With this in hand, we now note that $I[\lambda]$ is actually quite large.

Theorem (Shelah): Suppose that $\kappa^+<\sigma<\lambda$ for regular cardinals $\kappa,\sigma,\lambda$. Then there is a stationary $S\subseteq S^\lambda_\kappa$ in $I[\lambda]$ such that $S\cap \theta$ is stationary for stationarily-many $\theta\in S^\sigma_\kappa$.

Corollary: Suppose that $\kappa^{++}<\lambda$ for regular $\lambda,\kappa$. Then there is a stationary $S \subseteq S^\lambda_\kappa$ such that $S\in I[\lambda]$.

Thus, we see that a continuity condition is just a witness that these particular stationary sets live in $I[\lambda]$. Beyond giving us a link between squares and scales (which I want to fill out in the next post), $I[\lambda]$ is interesting in its own right. I won’t get into it much for now, but Todd Eisworth’s handbook chapter has a nice exposition on $I[\lambda]$ and its applications.

# Continuity Conditions and Exact Upper Bounds

I want to use this post to work through some material which solidifies the relationship between square-like principles and scales. We begin with a theorem due to Shelah.

Theorem: Let $\lambda, \kappa$ be regular cardinals with $\kappa^{++}<\lambda$. Then there is a stationary set $S\subseteq S^{\lambda}_{\kappa}$, and a sequence $\bar C=\langle C_\alpha : \alpha<\lambda\rangle$ such that:

1. Each $C_\alpha$ is a closed subset of $\alpha$;
2. If $\beta\in nacc(C_\alpha)$, then $C_\beta=C_\alpha\cap\beta$;
3. If $\delta\in S$, then $C_\delta$ is a club subset of $\delta$ with order-type $cf(\delta)=\kappa$.

The above sequence $\bar C$ is referred to as a continuity condition for $S$ by Shelah in Cardinal Arithmetic. We can think of these continuity conditions as weak fragments of square, which always hold in ZFC. The interesting thing is that continuity conditions allow us to produce exact upper bounds.

Definition: Let $\bar C=\langle C_\alpha : \alpha<\lambda$ be a continuity condition for some stationary set $S\subseteq S^\lambda_\kappa$ as above, and let $\vec f=\langle f_\alpha :\alpha<\lambda \rangle$ be a sequence of functions from $\kappa$ to the ordinals. We say $\vec f$ weakly obeys $\bar C$ if:

If $\alpha<\lambda$, then for each $\beta\in nacc(C_\alpha)$, we have $f_\beta(i) for each $i<\kappa$.

One thing to note is that despite having been published in 1994, Cardinal Arithmetic has no new material past 1989. In particular, the existence of continuity conditions over stationary sets was not known in the case that $\lambda=\mu^+$ for $\mu$ singular when cardinal arithmetic was sealed. However, Todd Eisworth’s chapter in the handbook has a nice exposition about the above theorem. On the other hand, the following is included in Cardinal Arithmetic (appearing as Claim 2.6A on page 16):

Theorem: Let $I$ be an ideal on a regular cardinal $\kappa$. 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 $\kappa$ to the ordinals;
3. $\vec f$ obeys $\bar C$.

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

Proof: We first produce a $\leq_I$-least upper bound, and then show that this bound must be exact. In order to produce the desired lub, we inductively produce better and better upper bounds $g_\xi$ which are better and better approximations to a lub as follows:

Stage $\xi=0$: Let $g_0$ be defined by $g_0(i)=\sup_{\alpha<\lambda} f_\alpha(i) + 1$.

Stage $\xi=\eta+1$: Suppose that $g_\eta$ has been defined, and is a $\leq_I$-upper bound for $\vec f$. If $g_\eta$ is a least-upper bound, then we can terminate the induction. Otherwise, there is some $g_{\eta+1}\leq_I g_\eta$ which is an upper bound for $\vec f$ such that $g_{\eta+1}\neq_I g_\eta$.

Stage $\xi$ limit: Suppose that $g_\eta$ has been defined for each $\eta<\xi$. For each $i<\kappa$, define the set $S_\xi(i)=\{g_\eta(i) : \eta<\xi\}$, and for each $\alpha<\lambda$, define $f^\xi_\alpha$ ,the projection of $f_\alpha$ to $S_\xi=\langle S_\xi(i) : i<\kappa$, by setting $f^\xi_\alpha(i)=\min(S_\xi(i)\setminus f_\alpha(i))$.

Note that each $f^\xi_\alpha$ is defined $I$-almost everywhere, and that the sequence $\vec{f}^\xi=\langle f^\xi_\alpha: \alpha<\lambda\rangle$ is $\leq_I$-increasing. We claim that there exists some $\alpha_\xi<\lambda$ such that for each $\alpha\in[\alpha_\xi, \lambda)$, we have $f^\xi_\alpha=_I f^\xi_{\alpha_\xi}$, and we set $g_\xi=f^\xi_\alpha$. To see this, assume otherwise.

As the sequence $\vec f^\xi$ never stabilizes we can find a club set $E\subseteq \lambda$ such that

$\alpha,\beta\in E$ and $\alpha<\beta$ implies $f^\xi_\alpha\neq_I f^\xi_\beta$.

Let $\delta\in acc(E)\cap S$, then $C_\delta\subseteq \delta$ is club in $\delta$ of order-type $\kappa^+$. We then inductively define an increasing sequence $\{\beta_\epsilon : \epsilon<\kappa^+\}$ of ordinals in $nacc( C_\delta)$ as follows:

We let $\beta_0\in nacc(C_\delta)$. If $\beta_\epsilon$ has been defined, then we can find $\beta_{\epsilon+1}\in nacc(C_\delta)$ be such that $f_{\beta_\epsilon}\neq_I f_{\beta_{\epsilon+1}}$. Just pick $\beta<\delta$ such that $\beta\in E$ and $\beta>\beta_\epsilon$ (which we can do since $\delta$ is an accumulation point of $E$), and let $\beta_{\epsilon+1}>\beta$ be in $nacc(C_\delta)$. At limit stages, let $\beta_{\epsilon}'=\sup_{\gamma<\epsilon}\beta_\gamma\in C_\delta$, and let $\beta_\epsilon>\beta_\epsilon'$ be in $nacc(C_\delta)$. Then our collection has the following properties:

1. $\beta_\epsilon\cap C_\delta=C_{\beta_\epsilon}$;
2. $f^\xi_{\beta_\epsilon}\neq_I f^\xi_{\beta_{\epsilon+1}}$.

Now for each $\epsilon<\kappa^+$, define the set

$t_\epsilon=\{i<\kappa : f^\xi_{\beta_\epsilon}(i).

By condition 2, each $t_\epsilon\notin I$, and so for each $\epsilon<\kappa^+$ we can pick $i(\epsilon)\in t_\epsilon$. Since we have $\kappa^+$-many of these sets, and $i$ ranges through $\kappa$, there must be some unbounded subset of $\kappa^+$ for which $i(\epsilon)$ is constant. Call this constant value $i(*)$. Now suppose $\epsilon(1)<\epsilon(2)<\kappa^+$ are such that $i(\epsilon(1))=i(\epsilon(2))=i(*)$, then $\beta_{\epsilon(1)+1}\in C_\delta\cap\beta_{\epsilon(2)}=C_{\beta_(\epsilon(2))}$ (in fact, it’s in $nacc(C_{\beta(\epsilon(2))}$. Since $\vec f$ obeys $\bar C$, we have that $f_{\beta_{\epsilon(1)+1}}(i(*)), and so $f^\xi_{\beta_{\epsilon(1)+1}}(i(*))\leq f^\xi_{\beta_{\epsilon(2)}}(i(*))$. Finally, since $i(\epsilon(1))=i(\epsilon(2))$ are both in $t_{\epsilon(1))}$ and $t_{\epsilon(2))}$ respectively, we get the following string of inequalities:

$f^\xi_{\beta_{\epsilon(1)}}(i(*)).

But then, the sequence $\langle f^\xi_{\beta_{\epsilon +1}}: i(\epsilon)=i(*)\rangle$ is strictly increasing along $S_\xi(i)$, which has size $\kappa$. Since $i(\epsilon)=i(*)$ happens $\kappa^+$-many times, this is absurd. Thus, the induction can be carried through limit stages.

Next, we claim that this induction cannot have been carried through $\kappa^+$-many stages. Otherwise, assume that $g_\xi$ has been defined for each $\xi<\kappa^+$, and let $\alpha(*)=\sup_{\xi<\kappa^+} \alpha_{\xi}<\lambda$. Next note since the sequence $\langle S_\xi (i) : \xi<\kappa^+$ is $\subseteq$-increasing for each $i<\kappa$, that the sequence $\langle f^\xi_{\alpha(*)} : \xi<\kappa^+\rangle$ must be decreasing and hence stabilize. Thus, we see that $\langle f^\xi_{\alpha(*)}: \xi<\kappa^+\rangle$ must also be eventually constant. On the other hand, $f^\xi_{\alpha(*)}=_I g_\xi$ for each $\xi<\kappa^+$, contradicting the fact that $g_\xi\neq_I g_\eta$ for $\xi,\eta$ large enough.

Therefore, $\vec f$ has a $\leq_I$-upper bound $g$. If we can show that $g$ is exact, then we are done. So suppose otherwise and let $h$ witness this. That is, $h$ is a function from $\kappa$ to the ordinals such that $h<_I g$, but there is no $\alpha<\lambda$ such that $h<_I f_\alpha$. So for each $\lambda$ define $t_\alpha=\{ i<\kappa : h(i)\leq f_\alpha(i)\}$, which is $I$-positive by assumption. Further, note that $\langle t_\alpha : \alpha<\lambda\rangle$ cannot stabilize modulo $I$ else if it does stabilize at some $\alpha(*)$, then define $F:\kappa\to ON$ by

$f(i)=h(i)$ if $i\in t_\alpha$ and $f(i)=g(i)$ otherwise.

It is straightforward to check that $F$ would then be an upper bound of $\vec f$ with $F\leq g$ and $F\neq_I g$, contradicting that $g$ is a lub. So as before, we can find a club $E\subseteq \lambda$ such that for all $\alpha<\beta$ in $E$, we have that $t_\alpha\neq t_\beta$ with $t_\beta\subseteq_I t_\alpha$. Letting $\delta\in S\cap acc(E)$, we can again find $\{\beta_\epsilon : \epsilon<\kappa^+\}$ such that

1. $\beta_\epsilon\cap C_\delta=C_{\beta_\epsilon}$;
2. $t_{\beta_\epsilon}\neq t_{\beta_{\epsilon +1}}$.

Let $i(\epsilon)\in t_{\beta_\epsilon}\setminus t_{\beta_{\epsilon+1}}$, and again find an unbounded subset of $\kappa^+$ such that $i(\epsilon)=i(*)$. If we let $\epsilon(1)<\epsilon(2)<\kappa^+$ be such that $i(\epsilon(1))=i(\epsilon(2))=i(*)$, we then get that

$h(i(*))\geq f_{\beta_{\epsilon(2)}}(i(*))>f_{\beta_{\epsilon(1)+1}} (i(*))> h(i(*))$.

This is silly, and so $g$ is an exact upper bound of $\vec f$.

# Coloring With Scales!

So the point of this post is to work through one of Shelah’s results that (roughly speaking) allows us to create a complicated coloring if we have a scale in hand. We begin with a few definitions.

Def: Let $\lambda, \kappa, \theta$ be cardinals with $\kappa+\theta<\lambda$. We say that $\mathrm{Pr}_1(\lambda, \kappa,\theta)$ holds if there is a coloring $[\lambda]^2\to\kappa$ of the pairs of $\lambda$ in $\kappa$-many colors such that:

Given a sequence $\langle t_\alpha : \alpha<\lambda\rangle$ of pairwise disjoint elements of $[\lambda]^{<\theta}$, and an ordinal $\zeta<\lambda$, there are $\alpha<\beta$ such that for every $\gamma\in t_\alpha$ and $\delta\in t_\beta$, we have $c(\{\gamma,\delta\})=\zeta$.

We can think of $\mathrm{Pr}_1(\lambda, \kappa,\theta)$ as a strong form of the assertion that $\lambda\not\to[\lambda]^2_\kappa$. For example, if we let $\mu$ be a singular cardinal, then $\mathrm{Pr}_1(\mu^+, \mu^+,2)$ says that there is a coloring $c:[\mu^+]^2\to \mu^+$ such that: Given any set $H\in[\mu^+]^{\mu^+}$ and any ordinal $\zeta<\mu$, there are $\alpha,\beta\in H$ such that $c(\{\alpha,\beta\})=\zeta$. In other words, this family of coloring principles assert that Ramsey’s theorem fails miserably, and in particular $\mathrm{Pr}_1(\mu^+, \mu^+,2)$ tells us that $\mu^+$ fails to be Jónsson in an incredibly spectacular way.

What I want to do in this post is work through a result of Shelah’s which allows us to produce witnesses to $\mathrm{Pr}_1(\mu^+, cf(\mu),cf(\mu))$ for any singular cardinal $\mu$ by using a scale to color pairs of $\mu^+$. First recall what a scale is.

Def: Let $\mu$ be singular. We say that the pair $(A,\vec f)$ is a scale on $\mu$ if:

1. $A$ is a set of regular cardinals cofinal in $\mu$;
2. $|A|<\min(A)$;
3. $\vec f=\langle f_\alpha : \alpha<\mu^+\rangle$ is strictly increasing and cofinal in $\prod A/J^{bd}[A]$ where $J^{bd}[A]$ is the ideal of bounded subsets of $A$.

A theorem of Shelah’s tells us that scales exist. Scales allow us to show that some of the combinatorial information on $\mu^+$ can be pulled down to $\mu$ or even $cf(\mu)$. In particular, scales are extremely useful for elementary submodel arguments because they allow us to smuggle information about $\mu^+$ into our submodel without needing all of $\mu$ or even $\mu^+$ present. This will be made apparent in the proof of the following (which can be found on page 67 of Cardinal Arithmetic):

Lemma 4.1B: (Shelah) Let $(A,\vec f)$ be a scale on $\mu$. Then, letting $\kappa=\min\{ |A\setminus a|: a\in A\}$ the principle $\mathrm{Pr}_1(\mu^+, \kappa,cf(|A|))$ holds. In particular, $\mathrm{Pr}_1(\mu^+, cf(\mu), cf(\mu))$ always holds.

The coloring used in the proof of the above lemma basically takes two ordinals $\alpha<\beta$ to the first point past which $f_\alpha$ and $f_\beta$ diverge completely. Much in the same way that coloring using Skolem functions is a useful technique, it’s nice to have this coloring lying around since scales always exist.

Proof: We begin by fixing a scale $(A,\vec f)$ as in the hypotheses, and let $\lambda=\mu^+$. Let $|A|=\theta$, and index $A=\{\mu_i : i<\theta\}$ so we may think of a function $f\in \prod A$ as a function in $\prod_{i<\theta}\alpha_i$. Now partition $A$ into $\kappa$-many pieces by way of $h: A\to \kappa$ such that $h^{-1}[\{\zeta\}]$ is unbounded in $\mu$ for each $\zeta<\kappa$. For ease of notation, let $A_\zeta=h^{-1}[\{\zeta\}]$. Now we define two colorings, $d,c$ on pairs of $\lambda$ as follows:

Let $d: [\lambda]^2\to\mu$ be defined by $d(\{\alpha,\beta\})=\sup\{i<\theta : f_\alpha(i)\geq f_\beta(i)\}$ for $\alpha<\beta$.

Let $c:[\lambda]^2\to\mu$ be defined by $c(\{\alpha,\beta\})=h(\mu_{d(\{\alpha,\beta\})})$. So $d$ takes a pair to the first point where their corresponding functions diverge, and $c$ takes the associated cardinal in $A$ to which piece of the partition that cardinal lies in. We claim that $c$ witnesses $\mathrm{Pr}_1(\lambda, \kappa,cf(\theta))$. With that said, fix a sequence $\langle t_\alpha : \alpha of pairwise disjoint elements of $[\lambda]^{ and note that there is some $\xi such that, for $\lambda$-many $\alpha<\lambda$, we have $t_\alpha=\{ t_\alpha^\eta : \eta<\xi\}$. So, we may as well assume that each $t_\alpha$ has this property by thinning out the collection we were originally handed, and reindexing. Further, let $\zeta<\kappa$ be a given color.

We want to show that there are $\alpha<\beta<\lambda$ such that for each $\eta_1,\eta_2<\xi$ we have $c(\{t_\alpha^{\eta_1},t_\beta^{\eta_2}\})=\zeta$. There are going to be a lot of indices to keep track of, but keeping the goal in mind (no matter how horrible it looks) will help. Our main tool for proving that this coloring is appropriate will be elementary submodels, so let $\xi$ be regular, and large enough with $M\prec \mathfrak{A}=(H(\theta),\in,<_\theta)$ such that:

1. $\{\lambda, \zeta, \langle t_\alpha : \alpha<\lambda\rangle, \vec f\}\subseteq M$;
2.  $\zeta\cup A\subseteq M$;
3. $|M|<\mu$.

It won’t quite be apparent why we’re utilizing this machinery until a little bit later. Now since the collection $\langle t_\alpha : \alpha<\lambda\rangle$ is disjoint, we can assume that $t_\alpha^\eta\geq\alpha$ for each $\alpha<\lambda$ by reindexing the sets $t_\alpha$, and perhaps chopping off initial segments. Next let $A'=\{\mu_i\in A : \sup (M\cap \mu_i)<\mu_i\}$ and define the characteristic function of $M$ with respect to $A$ by $Ch^A_M(i)=\sup (M\cap \mu_i)$ for every $\mu_i\in A'$ and $0$ everywhere else. Since $|M\cap \lambda|<\mu$, it follows that $Ch^A_M$ is non-zero almost everywhere (mod bounded) and so is in $\prod A/J^{bd}[A]$. Next since $\vec f$ is cofinal in $\prod A/J^{bd}[A]$, there is some $\beta(0)<\lambda$ such that:

1. $Ch^A_M <_{J^{bd}[A]}f_{\beta(0)}$;
2. $\beta(0)>\sup (M\cap \lambda)$.

So not only does $f_{\beta(0)}(i)$ get above $\sup (M\cap \mu_i)$ almost everywhere, we have that $\beta(0)> \sup (M\cap \lambda)$. This will allow us to take an appropriate Skolem Hull of $M$ and know that these things still happen. Next, since $t^\eta_{\beta(0)}\geq\beta(0)$, we can fix for each $\eta<\xi$ an index $i_eta$ such that: For every $j\geq i_\eta$, we have that $Ch^A_M(j). Let $i(0)=\sup\{i_\eta : \eta<\xi\}<\theta$ since $\xi, so then for every $j> i(0)$ and every $\eta<\xi$, we have $Ch^A_M(j). This give us a pair of canonical witnesses to the fact that $M\cap\lambda$ is thin, and gives us one of the indices we will use to witness the desired coloring property. So our next goal is to find our second (smaller) witness, which we will want $M$ to see enough of.

For each $\alpha<\lambda$, define a function $f_\alpha^*\in\prod \mu_i$ by:

$f_\alpha^*(i)=\min\{ f_{t^\eta_\alpha}(i) : \eta<\xi\}$.

Note that since $t^\eta_\alpha\geq\alpha$, we have that $f_\alpha\leq_{J^{bd}[A]} f^*_\alpha$. So above each $f_\alpha$, we have $\xi$-many functions $f_{t^\eta_\alpha}$ above it, and we are letting $f_\alpha^*(i)$ be the closest approximation to $f_\alpha$ we can get by way of these ladders of functions. Next let $A^*=\{\mu_i\in A : \sup\{f_\alpha^*(i): \alpha<\lambda\}=\mu_i\}$, which we claim is $J^bd[A]$-equivalent to $A$. Otherwise, suppose that $A\setminus A^*\notin J^{bd}[A]$, and define a function $g\in \prod A$ by $g(i)= \sup\{f_\alpha^*(i): \alpha<\lambda\}$ if $\mu_i\in A\setminus A^*$ and $0$ otherwise. As before, we can find some $\alpha<\lambda$ such that $0<_{J^{bd}[A]}g<_{J^{bd}[A]} f_\alpha\ \leq_{J^{bd}[A]} f^*_\alpha$ which is absurd. Now we pick an index $i(1)<\xi$ such that:

1. $i(1)\in A_\zeta\cap A^*$;
2. $i(1)>i(0)$;
3. $A\setminus \mu_{i(1)}\subseteq A'$;
4. $i(1)>|M|$.

We now make our first approximation to our desired companion to $\beta(0)$. We let $\beta(1)$ be such that

$\tau:= f^*_{\beta(1)} (i(1))>\sup\{f_{t^\eta_{\beta(0)}}(i(1)) : \eta<\xi\}$

We note that such a choice is possible, as $a(1)\in A^*$ implies that $\sup\{f_\alpha^*(i(1)):\alpha<\lambda\}=\mu_{i(1)}$ whereas $\sup\{f_{t^\eta_{\beta(0)}}(i(1)) : \eta<\xi\}<\mu_{i(1)}$ since $\xi. Really, we only wanted to find an index such that we can get above each $f_{t^\eta_{\beta(0)}}(i(1))$, and the important thing here is $\tau$. With that said, let $N=Sk^{\mathfrak{A}}(M\cup\{\tau\})$, and note that we have $\mathfrak{A}\models (\exists \alpha<\lambda)(f_\alpha^*(i(1))=\tau)$. So by elementarity we have

$N\models (\exists \alpha<\lambda)(f_\alpha^*(i(1))=\tau)$ by our requirements on $M$, and the fact that $\tau\in N$. So let $\beta(2)\in N\cap\lambda$ be such that $f_{\beta(2)}^*((i(1))=\tau$. Thus:

$\eta_1,\eta_2<\xi\implies f_{t^{\eta_1}_{\beta(0)}}(i(1))<\tau=f^*_{\beta(2)}(i(1))\leq f_{t^{\eta_2}_{\beta(2)}}(i(1))$.

We claim that $\beta(0)$ and $\beta(2)$ are as desired. That is, we need to show that $\beta(2)<\beta(0)$ and for every pair $\eta_1,\eta_2$, we have $c(\{t^{\eta_1}_{\beta(2)},t^{\eta_2}_{\beta(0)}\})=\zeta$ (so also that $t^{\eta_1}_{\beta(2)}). So if we can show that $\sup \{i<\theta : f_{t^{\eta_1}_{\beta(2)}}(i)\geq f_{t^{\eta_2}_{\beta(0)}}(i)\}=i(1)$, then we are done since $i(1)\in A_\zeta$. But by the above inequality, we only have to show the following three things:

1. $\beta(2)<\beta(0)$;
2. For every pair $\eta_1,\eta_2$, we have $t^{\eta_1}_{\beta(2)};
3. For every $j>i(1)$, we get that $f_{t^{\eta_1}_{\beta(2)}}(j).

By a lemma due to Baumgartner,  we have that (by our choice of $i(1)$, for every regular $\sigma \in M$, if $\sigma>\mu_{i(1)}$, then

$\sup (M\cap \sigma)=\sup(N\cap\sigma)$.

Now since $\lambda>\mu_{i(1)}$, it follows since $\beta(0)>\sup (M\cap \lambda)$, that $\beta(0)>\sup(N\cap\lambda)$ as well. But since $\beta(2)\in N\cap\lambda$, we have that $\beta(0)>\beta(2)$, and similarly $t^{\eta_1}_{\beta(2)}<\beta(0)\leq t^{\eta_2}_{\beta(0)}$ for each $\eta_1,\eta_2<\xi$ since each $t^{\eta_1}_{\beta(2)}$ is definable from parameters in $N$. Now that we’ve shown 1. and 2. above, we only need to show 3.

Now for every $j\geq i(1)$, and $\eta_1,\eta_2<\xi$ we have that $f_{t^{\eta_1}_{\beta(2)}}(j)\in N$ since it’s definable from parameters, and hence $f_{t^{\eta_1}_{\beta(2)}}(j). On the other hand, our choice of $\beta(0)$ tells us that $Ch^A_m(j) < f_{t^{\eta_2}_{\beta(0)}}(j)$, and so the result follows.

# 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 $I$.

# An Interlude and Some Motivation (Sh506)

I’m interested in the following open question:

Is there a singular cardinal $\mu$ such that $\mu^+$ is a Jónsson cardinal?

I think all of the projects that I’m working on are somehow related to the above question. In particular, that’s been my motivation for working through Sh506. So, I figured I would take some space to make that motivation precise. We’re going to start with a couple of definitions:

Definition: Suppose that $I$ is an ideal over some cardinal $\kappa$. We say that $I$ is $\theta$-indecomposable for a regular cardinal $\theta$ if, for every $I$-positive set $A$ and every function $f:A\to \theta$, there exists a $\rho<\theta$ such that $f^{-1}[\{\rho\}]$ is $I$-positive.

Definition: We say that $I$ is $\theta$-weakly saturated if any collection $\{A_i : i<\theta\}$ of $\theta$-many $I$-positive sets cannot be disjoint.

Both of these properties say something about our ability to split up sets with respect to my ideal. Indecomposability says that partitioning a positive set into many pieces yields a positive set in one of the cells. On the other hand, weak saturation tells me that I cannot partition $\kappa$ into many medium-sized pieces. The conjunction of these two statements turns out to be somewhat powerful. The following result appears as Theorem 2 of This paper by Todd Eisworth :

Theorem(Eisworth): Suppose that $\kappa$ is a cardinal with $I$ an ideal over $\kappa$ with the property that there is some regular $\theta<\kappa$ such that $I$ is $\theta$-indecomposable and $\theta$-weakly saturated. Then if $\lambda$ is the completeness of $I$, there is a stationary set $S\subseteq\kappa$ such that every sequence $\langle T_i : i<\xi \rangle$ of stationary subsets of $S$ with $\xi<\lambda$ reflects simultaneously.

We then have the following result due to Shelah:

Theorem(Shelah): If $\mu$ is a singular cardinal with $\mu^+$ Jónsson, then there is a regular cardinal $\theta<\mu$, and an ideal $I$ which is $cf(\mu)$-complete, $\theta$-indecomposable, and $\theta$-weakly saturated.

Now, it’s known that the existence of such an ideal is consistent at $\mu^+$ for $\mu$ a singular cardinal of countable cofinality. This note works through the proof of this. It was also shown in the paper by Eisworth that, if $I$ is an ideal which is $\theta$-weakly saturated and $\theta$-indecomposable, then any collection of $\theta$-many $I$-positive sets has an unbounded subcollection with non-empty intersection. Say that the ideal $I$ satisfies $(*)_\theta$ if this holds. Suppose now that $\mu$ is a singular cardinal and $\mu^+$ carries an ideal satisfying $(*)_\theta$. Define a map

$\pi: \mu^+\to \mu$ given by $\pi(\alpha)=cf(\alpha)$.

Let $J$ be the Rudin-Keisler projection of $I$ to $\mu$ by way of $\pi$. Then we see that $J$ also satisfies $(*)_\theta$ (also, $J$ is $\theta$-weakly saturated). If $\mu^+$ is Jónsson, a theorem of Shelah’s tells us that $\mu$ must be the limit of inaccessible cardinals, and hence an $\aleph$-fixed point. So, we can try to do pcf theory at $\mu$ with this ideal $J$ that we have. Now, I also mentioned in the previous post that I suspect that the trichotomy theorem holds under our original assumptions. I was unable to make it go through with just weak saturation, but it does hold if we assume the ideal we are working over satisfies $(*)_\theta$. So in the next blog entry, I plan on showing this (and then some). So now that we have trichotomy and $\lambda$-directedness of $\prod A/J_{<\lambda}$, many of the big pcf theorems will go through pretty easily. In particular, we will be able to show that the pcf theorem holds in this case.

I do want to point out though, that Shelah has a proof of the pcf theorem in Sh506 without assuming $(*)_\theta$ which makes our approach suboptimal. However, the proof we will give is cleaner, and is basically the same as the ones from the Abraham-Magidor chapter in the Handbook of Set Theory. Further, my motivation is wanting to do pcf theory at singular cardinals $\mu$ such that $\mu^+$ is Jónsson, so it all works out.

Anyway, my plan future entries is to work through:

The existence of universal cofinal sequences (assuming just weak saturation);

The Trichotomy Theorem (assuming $(*)_\theta$);

The pcf theorem (assuming $(*)_\theta$).