Set theory seminar – Marion Scheepers: Coding strategies (IV)

For the third talk (and a link to the second one), see here. The fourth talk took place on October 12.

We want to show the following version of Theorem 2:

Theorem. Suppose $\kappa$ is a singular strong limit cardinal of uncountable cofinality. Then the following are equivalent:

For each ideal $J$ on $\kappa$ , player II has a winning coding strategy in $RG(J)$ .

$2^\kappa<\kappa^{+\omega}$ .

Since $2^\kappa$ has uncountable cofinality, option 2 above is equivalent to saying that the instance of ${\sf wSCH}$ corresponding to $\kappa$ holds.

Before we begin the proof, we need to single out some elementary consequences in cardinal arithmetic of the assumptions on $\kappa$ . First of all, since $\kappa$ is singular strong limit, then for any cardinal $\lambda<\kappa$ , we have that

$\kappa^\lambda=\left\{\begin{array}{cl}\kappa&\mbox{\ if }\lambda<{\rm cf}(\kappa),\\ 2^\kappa&\mbox{\ otherwise.}\end{array}\right.$

Also, since the cofinality of $\kappa$ is uncountable, we have Hausdoff’s result that if $n<\omega$ , then $(\kappa^{+n})^{\aleph_0}=\kappa^{+n}$ . I have addressed both these computations in my lecture notes for Topics in Set Theory, see here and here.

We are ready to address the Theorem.

Proof. $(2.\Rightarrow 1.)$ We use Theorem 1. If option 1. fails, then there is an ideal $J$ on $\kappa$ with ${\rm cf}(\left< J\right>,{\subset})>|J|$ .

Note that ${\rm cf}(\left< J\right>,{\subset})\le({\rm cf}(J,{\subset}))^{\aleph_0}$ , and $\kappa\le|J|$ . Moreover, if $\lambda<\kappa$ , then $2^\lambda<{\rm cf}(J,{\subset})$ since, otherwise,

$({\rm cf}(J,{\subset}))^{\aleph_0}\le 2^{\lambda\aleph_0}=2^\lambda<\kappa$ .

So ${\rm cf}(J,{\subset})\ge\kappa$ and then, by Hausdorff, in fact ${\rm cf}(J,{\subset})\ge \kappa^{+\omega}$ , and option 2. fails.

$(1.\Rightarrow 2.)$ Suppose option 2. fails and let $\lambda=\kappa^{+\omega}$ , so $\kappa<\lambda<2^\kappa$ and ${\rm cf}(\lambda)=\omega$ . We use $\lambda$ to build an ideal $J$ on $\kappa$ with ${\rm cf}(\left< J\right>,{\subset})>|J|$ .

For this, we use that there is a large almost disjoint family of functions from ${\rm cf}(\kappa)$ into $\kappa$ . Specifically:

Lemma. If $\kappa$ is singular strong limit, there is a family ${\mathcal F}\subseteq{}^{{\rm cf}(\kappa)}\kappa$ with ${}|{\mathcal F}|=2^\kappa$ and such that for all distinct $f,g\in{\mathcal F}$ , we have that ${}|\{\alpha<{\rm cf}(\kappa)\mid f(\alpha)=g(\alpha)|<{\rm cf}(\kappa)$ .

In my notes, I have a proof of a general version of this result, due to Galvin and Hajnal, see Lemma 12 here; essentially, we list all functions $f:{\rm cf}(\kappa)\to\kappa$ , and then replace them with (appropriate codes for) the branches they determine through the tree $\kappa^{{\rm cf}(\kappa)}$ . Distinct branches eventually diverge, and this translates into the corresponding functions being almost disjoint.

Pick a family ${\mathcal F}$ as in the lemma, and let ${\mathcal G}$ be a subfamily of size $\lambda$ . Let $S=\bigcup{\mathcal G}\subseteq{\rm cf}(\kappa)\times\kappa$ . We proceed to show that $|S|=\kappa$ and use ${\mathcal G}$ to define an ideal $J$ on $S$ as required.

First, obviously $|S|\le\kappa$ . Since $\kappa<\lambda=|{\mathcal G}|$ and ${\mathcal G}\subseteq{\mathcal P}(S)$ , it follows that ${}|S|\ge\kappa$ , or else ${}|{\mathcal P}(S)|<\kappa$ , since $\kappa$ is strong limit.

Now define

$J=\{X\subseteq S\mid\exists {\mathcal H}\subseteq{\mathcal G}\,(|{\mathcal H}|<\omega,\bigcup{\mathcal H}\supseteq X)\}.$

Clearly, $J$ is an ideal. We claim that $|J|=\lambda$ . First, each singleton $\{f\}$ with $f\in{\mathcal G}$ is in $J$ , so ${}|J|\ge\lambda$ . Define $\Phi:[{\mathcal G}]^{<\aleph_0}\to J$ by $\Phi({\mathcal H})=\bigcup{\mathcal H})$ . Since the functions in ${\mathcal G}$ are almost disjoint, it follows that $\Phi$ is 1-1. Let $G$ be the image of $\Phi$ . By construction, $G$ is cofinal in $J$ . But then

${}|J|\le|{\mathcal G}|2^{{\rm cf}(\kappa)}=\lambda 2^{{\rm cf}(\kappa)}=\lambda$ ,

where the first inequality follows from noticing that any $X\in J$ has size at most ${\rm cf}(\kappa)$ . It follows that $|J|=\lambda$ , as claimed.

Finally, we argue that ${\rm cf}(\left< J\right>,{\subset})>\lambda$ , which completes the proof. For this, consider a cofinal ${\mathcal A}\subseteq\left< J\right>$ , and a map $f:{\mathcal A}\to[{\mathcal G}]^{\le\aleph_0}$ such that for all $A\in{\mathcal A}$ , we have $A\subseteq\bigcup f(A)$ .

Since ${\mathcal A}$ is cofinal in $\left< J\right>$ , it follows that $f[{\mathcal A}]$ is cofinal in ${}[{\mathcal G}]^{\le\aleph_0}$ . But this gives the result, because

${}|{\mathcal A}|\ge{\rm cf}([{\mathcal G}]^{\le \aleph_0},{\subset})={\rm cf}([\lambda]^{\le \aleph_0},{\subset})>\lambda$ ,

and we are done. $\Box$

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One Response to Set theory seminar – Marion Scheepers: Coding strategies (IV)

Set theory seminar – Marion Scheepers: Coding strategies (III) « A kind of library says:

August 28, 2012 at 12:13 pm

[…] the assumption, and trying to address it leads to Theorem 2, which will be the subject of the next (and last) talk. 43.614000 -116.202000 Like this:LikeBe the first to like […]

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