## [Sh:430.1-4] An easy inequality

January 31, 2011 at 16:38 | Posted in Cardinal Arithmetic, cov vs. pp, [Sh:355], [Sh:430] | 1 Comment

What follows is not really part of the first section of [Sh:430], but it does pin down an important connection between cov and pp. The proposition is a special case of part of Shelah’s “cov vs. pp theorem”, Theorem 5.4 on page 87 of Cardinal Arithmetic. Note that there are no special assumptions on ${\mu}$ here. Getting inequalities in the reverse direction (i.e., showing that covering numbers are less than pp numbers) is generally a more difficult proposition.

Let us recall that ${{\rm cov}(\mu,\mu,\kappa^+, 2)}$ is the minimum cardinality of a set ${\mathcal{P}\subseteq[\mu]^{<\mu}}$ such that for any ${A\in [\mu]^\kappa}$ (equivalently ${[\mu]^{\leq\kappa}}$), there is a ${B\in\mathcal{P}}$ such that ${A\subseteq B}$.

Proposition 1 If ${\mu}$ is singular of cofinality ${\kappa}$, then

$\displaystyle {\rm pp}(\mu)\leq {\rm cov}(\mu,\mu,\kappa^+,2). \ \ \ \ \ (1)$

Proof: Suppose by way of contradiction that

$\displaystyle \theta:={\rm cov}(\mu,\mu,\kappa^+,2)<{\rm pp}(\mu). \ \ \ \ \ (2)$

By the definition of ${\rm pp}$, we can find a (not necessarily increasing) sequence of regular cardinals ${\langle \mu_i:i<\kappa\rangle}$ and an ultrafilter ${D}$ on ${\kappa}$ such that

$\displaystyle \{i<\kappa:\mu_i\leq\tau\}\notin D\text{ for each }\tau<\mu, \ \ \ \ \ (3)$

and

$\displaystyle \theta<\sigma:={\rm cf}\left(\prod_{i<\kappa}\mu_i, <_D\right). \ \ \ \ \ (4)$

Let ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ be a family of cardinality ${\theta}$ standing as witness to ${{\rm cov}(\mu,\mu,\kappa^+,2)=\theta}$, and let ${\langle f_\alpha:\alpha<\sigma\rangle}$ be a ${<_D}$-increasing and cofinal sequence of functions in ${\prod_{i<\kappa}\mu_i}$. For each ${\alpha<\sigma}$, the range of ${f_\alpha}$ is a subset of ${\mu}$ of cardinality at most ${\kappa}$, and so we can find ${A_\alpha\in \mathcal{P}}$ such that

$\displaystyle {\rm ran}(f_\alpha)\subseteq A_\alpha. \ \ \ \ \ (5)$

Since ${\sigma}$ is a regular cardinal greater than ${\theta}$, there is a single ${A^*\in \mathcal{P}}$ such that

$\displaystyle |\{\alpha<\sigma: A_\alpha=A^*\}|=\sigma. \ \ \ \ \ (6)$

Thus, (by passing to a subsequence of ${\langle f_\alpha:\alpha<\sigma\rangle}$) we may as well assume that the range of each ${f_\alpha}$ is a subset of ${A^*}$.

But ${|A^*|<\mu}$, and so

$\displaystyle B:= \{i<\kappa:|A^*|<\mu_i\}\in D \ \ \ \ \ (7)$

by our choice of ${D}$. Let us now define a function ${g}$ by setting ${g(i)=0}$ if ${i\notin B}$, and

$\displaystyle g(i)=\sup(A^*\cap\mu_i) \ \ \ \ \ (8)$

whenever ${i}$ is in ${B}$. Our assumptions imply that ${g}$ is in ${\prod_{i<\kappa}\mu_i}$.

If ${\alpha<\sigma}$, then for each ${i\in B}$ we have

$\displaystyle f_\alpha(i)\leq g(i) \ \ \ \ \ (9)$

as ${f_\alpha(i)\in A^*\cap\mu_i}$.

It follows that ${g}$ is a function in ${\prod_{i<\kappa}\mu_i}$ such that${f_\alpha\leq_D g}$ for all ${\alpha<\sigma}$. This is absurd, given our choice of ${\langle f_\alpha:\alpha<\sigma\rangle}$, and the proof is complete. $\Box$

[Updated 2-7-11]

## [Sh:430.1-3] Elementary Submodel Conventions

We’re going to be working with elementary submodels a lot, so I wanted to pin down the assumptions we make and the notation we use.

In the current project, we are investing combinatorics relevant to a singular cardinal ${\mu}$, and ${\chi}$ is a regular cardinal “much much larger” than ${\mu}$. The exact identity of ${\chi}$ isn’t important. We also need to fix some well-ordering ${<_\chi}$ of ${H(\chi)}$; we will build this into the structures we consider so that they all have definable Skolem functions.

In general, ${\mathfrak{A}}$ denotes some expansion of ${\langle H(\chi),\in, <_\chi\rangle}$ by at most countably many functions, constants, and relations. I won’t get too pedantic with this, but for example, in the current situation we want ${\mathfrak{A}}$ to contain a name for ${\mu}$, so that we know ${\mu}$ makes it into any elementary submodel of ${\mathfrak{A}}$. We won’t make this explicit in our notation.

I will use ${{\rm Sk}_{\mathfrak{A}}(B)}$ (or just ${{\rm Sk}(B)}$) to denote the Skolem hull of ${B}$ in the structure ${\mathfrak{A}}$.

I may update this post to reflect new things that I forgot to include…

## [Sh:430.1-2] Framework for proof

Let us assume that ${\mu}$ is a singular cardinal with ${\rm{pp}(\mu)=\mu^+}$. We will prove that there is a family ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ of cardinality ${\mu^+}$ such that every member of ${[\mu]^{\rm{cf}(\mu)}}$ is a subset of some member of ${\mathcal{P}}$.

Fix a sufficiently large regular cardinal ${\chi}$; we will be working with elementary submodels of the structure ${\langle H(\chi),\in, <_\chi\rangle}$ where ${<_\chi}$ is some appropriate well-ordering of ${H(\chi)}$ used to give us definable Skolem functions.

Let ${\mathfrak{M}=\langle M_\alpha:\alpha<\mu^+\rangle}$ be a ${\mu^+}$-approximating sequence, that is, ${\mathfrak{M}}$ is a ${\in}$-increasing and continuous sequence of elementary submodels of ${H(\chi)}$ such that for each ${\alpha<\mu^+}$,

• ${\mu\in M_0}$,
• ${M_\alpha}$ has cardinality ${\mu}$,
• ${M_\alpha\cap\mu^+}$ is an initial segment of ${\mu^+}$ (so ${\mu+1\subseteq M_\alpha\cap\mu^+}$), and
• ${\langle M_\beta:\beta<\alpha\rangle\in M_{\alpha+1}}$.

Let ${M^*}$ denote ${\bigcup_{\alpha<\mu^+} M_\alpha}$. Then ${M^*}$ is an elementary submodel of ${H(\chi)}$ of cardinality ${\mu^+}$ containing ${\mu^+}$ as both an element and subset. Define

$\displaystyle \mathcal{P}=M^*\cap [\mu]^{<\mu}. \ \ \ \ \ (1)$

Clearly ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ and ${|\mathcal{P}|=\mu^+}$, so we need only verify that for any ${a\in [\mu]^{\rm{cf}(\mu)}}$, there is a ${b\in \mathcal{P}}$ with ${a\subseteq b}$.

In broad terms, this will be done via an “${I[\lambda]}$ argument” with ${\lambda=\mu^+}$, but we’ll fill in the details in further posts.

## [Sh:430.1-1] What are we aiming for?

We focus our attention on the first section of [Sh:430]: “equivalence of two covering properties”. The first result states the following:

Theorem 1 If ${\rm{pp}(\lambda)=\lambda^+}$, ${\lambda>\rm{cf}(\lambda)=\kappa>\aleph_0}$ then ${\rm{cov}(\lambda,\lambda,\kappa^+,2)=\lambda^+}$.

I am pretty sure that this isn’t what Shelah meant to write here — the assumption that ${\lambda}$ has uncountable cofinality means that the result is a trivial consequence of a deeper theorem in Cardinal Arithmetic [Update: Perhaps not! See 2nd update below]. I think that the proof he gives works even for the countable cofinality case, and this gives us something of interest because ${\lambda}$ could very well be a fixed point. So, here’s what I conjecture he meant to say (after changing notation and translating the “cov” statement into more standard form):

Theorem 2 Let ${\mu}$ be a singular cardinal. Then ${\rm{pp}(\mu)=\mu^+}$ if and only if there is a family ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ of cardinality ${\mu^+}$ such that every member of ${[\mu]^{\rm{cf}(\mu)}}$ is a subset of some element of ${\mathcal{P}}$.

We’ll work through his proof as best we can, and see if my conjecture is correct.

UPDATE 1: I don’t think the conjecture is correct. It looks to me like the proof was originally written for singular of countable cofinality, but a mistake was discovered and the statement was corrected but a lot of the old proof didn’t get revised properly. The annotated content was revised, but the abstract was not. Anyway, I’ll ask Saharon if I can’t figure out what’s going on.

UPDATE 2: I think the theorem doesn’t follow from the Cardinal Arithmetic stuff. The problem I run into is that when Shelah says informally that “cov = pp when the cardinal has uncountable cofinality”, there are lots of disclaimers hidden in the background. But this underscores why I think what I’m doing here is important — I want to pin down what exactly is known and what exactly is still open in this area.

## Current Project: [Sh:430] Section 1

So, I’ve decided what I want to try to accomplish with this blog: I want to use it as a tool for developing a reasonable exposition of more advanced pcf material. I consider this an experiment, as I have no idea how effective this will be, and it is entirely possible that I decide the entire endeavour is silly.

My first project will be to develop a treatment of the first section of [Sh:430] “Further Cardinal Arithmetic”; my intermediate goal is to concentrate on resuts related to the “cov vs. pp” problem.

It is safe to say that most of what I will be doing is presenting results of Shelah; if something is unattributed then consider it a result of Shelah (although I tend to try and reformulate things to make them more understandable to myself).

## SCH above a compact cardinal

January 20, 2011 at 20:41 | Posted in Uncategorized | 1 Comment
Tags: ,

This is really just an experiment to see if I can actually get latex2wp to work for me, but maybe it has mathematical interest as well. Anyway, it is well-known that Shelah’s pcf theory can be used to give a proof of Solovay’s result that the Singular Cardinals Hypothesis holds above a compact cardinal. What follows is a fairly short proof of a more general result; the argument consists of chaining together some ideas of Shelah taken from his Cardinal Arithmetic.
See “proof of 3.3 in Case gamma” and Remark 3.3B on pages 150-151 of his book.

Theorem 1 Suppose ${\mu}$ is singular of countable cofinality and ${\rm{pp}(\mu)>\mu^+}$. Then there is no uniform countably complete ultrafilter on ${\mu^+}$.

Proof: We actually show more: we prove that there is no weakly ${\mu^+}$-saturated uniform countably complete filter on ${\mu^+}$. Said another way, if ${F}$ is a uniform countably complete filter on ${\mu^+}$ for ${\mu}$ singular of countable cofinality and ${\text{pp}(\mu)>\mu^+}$, then we can find ${\mu^+}$ disjoint ${F}$-positive subsets of ${\mu^+}$. In particular, ${F}$ cannot be an ultrafilter.

To understand the proof, we don’t really need to know what ${\text{pp}(\mu)>\mu^+}$ means. What we really need is one of the combinatorial consequences of this statement, namely that there is a family ${\langle A_\alpha:\alpha<\mu^+\rangle}$ of countable subsets of ${\mu}$ (not ${\mu^+}$!) such that for every ${\beta<\alpha}$, the collection ${\langle A_\beta:\beta<\alpha\rangle}$ is essentially disjoint, in the sense that there is a function ${F_\alpha}$ with domain ${\alpha}$ such that

• ${F_\alpha(\beta)}$ is a finite subset of ${A_\beta}$, and
• the collection ${\langle A_\beta\setminus F_\alpha(\beta):\beta<\alpha\rangle}$ is pairwise disjoint.

Thus, for each ${\alpha<\mu^+}$ the collection ${\langle A_\beta:\beta<\alpha\rangle}$ can be “disjointified” by removing a finite subset from each ${A_\beta}$.

Note that since the ${A_\alpha}$ are subsets of ${\mu}$, no ${\mu^+}$-sized subcollection of ${\langle A_\alpha:\alpha<\mu^+\rangle}$ can be disjointified, as this would give us ${\mu^+}$ disjoint subsets of ${\mu}$. One should think of ${\langle A_\alpha:\alpha<\mu^+\rangle}$ as a failure of “compactness” at ${\mu^+}$ which necessarily arises from the failure of the singular cardinals hypothesis at ${\mu}$.

Let us assume now that ${\langle A_\alpha:\alpha<\mu^+\rangle}$ is as above, and ${F}$ is a uniform countably complete filter on our cardinal ${\mu^+}$; we will show ${\mu^+}$ contains ${\mu^+}$ disjoint ${F}$-positive subsets. Let us also assume that we have fixed some way of enumerating each ${A_\alpha}$, so that expressions such as “the first 5 elements of ${A_\alpha}$” make sense.

Given ${\beta<\mu^+}$ and ${n<\omega}$, define ${X^\beta(n)}$ to be those ${\alpha<\mu^+}$ greater than ${\beta}$ for which ${F_\alpha(\beta)}$ is contained in the first ${n}$ elements of ${A_\beta}$. For each ${\beta}$, the sequence ${\langle X^\beta(n):n<\omega\rangle}$ is increasing with union ${(\beta,\mu^+)}$. Since ${F}$ is uniform and countably complete, it follows that there is an ${n_\beta<\omega}$ such that ${X^\beta(n_\beta)}$ is ${F}$-positive.

For each ${\beta<\mu^+}$, let ${x_\beta}$ be the “${n_\beta+1}$th element” of ${A_\beta}$. Then clearly the set

$\displaystyle Y_\beta:=\{\alpha<\mu^+:\beta<\alpha\text{ and }x_\beta\in A_\beta\setminus F_\alpha(\beta)\} \ \ \ \ \ (1)$

is ${F}$-positive for each ${\beta}$, as includes ${X^\beta(n_\beta)}$.

We know ${x_\beta<\mu}$ for each ${\beta}$, and so there exists an ${x^*<\mu}$ such that

$\displaystyle |\{\beta<\mu^+:x_\beta=x^*\}|=\mu^+. \ \ \ \ \ (2)$

Given ${x^*}$, we define

$\displaystyle Z:=\{\beta<\mu^+:x_\beta=x^*\}. \ \ \ \ \ (3)$

We are now done, as given ${\alpha<\beta}$ in ${Z}$, we see that ${Y_\alpha\cap Y_\beta=\emptyset}$, and therefore the collection ${\{Y_\alpha:\alpha\in A\}}$ is a collection of ${\mu^+}$ disjoint ${F}$-positive subsets of ${\mu^+}$. $\Box$

Corollary 2 If ${\kappa<\mu}$ where ${\kappa}$ is strongly compact and ${\mu}$ is a singular cardinal of countable cofinality, then ${\text{pp}(\mu)=\mu^+}$.

Proof: The co-bounded filter on ${\mu^+}$ is a uniform ${\mu^+}$-complete filter. Since ${\kappa}$ is strongly compact, this filter can be extended to a ${\kappa}$-complete ultrafilter (necessarily uniform) on ${\mu^+}$. The conclusion now follows immediately from the theorem. $\Box$

Corollary 3 If ${\kappa}$ is strongly compact, then the Singular Cardinals Hypothesis holds above ${\kappa}$.