## Set theory seminar – Marion Scheepers: Coding strategies (I)

This semester, the seminar started with a series of talks by Marion. The first talk happened on September 14.

We consider two games relative to a (proper) ideal $J\subset{\mathcal P}(S)$ for some set $S$. The ideal $J$ is not assumed to be $\sigma$-complete; we denote by $\left< J\right>$ its $\sigma$-closure, i.e., the collection of countable unions of elements of $J$. Note that $\left< J\right>$ is a $\sigma$-ideal iff $\left< J\right>$ is an ideal iff $S\notin\left< J\right>$.

The two games we concentrate on are the Random Game on $J$, $RG(J)$, and the Weakly Monotonic game on $J$, $WMG(J)$.

In both games, players I and II alternate for $\omega$ many innings, with I moving first, moving as follows:

$\begin{array}{cccccc} I&O_0\in\left< J\right>&&O_1\in\left< J_2\right>&&\cdots\\ II&&T_0\in J&&T_1\in J \end{array}$

In $RG(J)$ we do not require that the $O_i$ relate to one another in any particular manner (thus “random”), while in $WMG(J)$ we require that $O_1\subseteq O_2\subseteq\dots$ (thus “weakly”, since we allow equality to occur).

In both games, player II wins iff $\bigcup_n T_n\supseteq\bigcup_n O_n$. Obviously, II has a (perfect information) winning strategy, with $=$ rather than the weaker $\supseteq$.

However, we are interested in an apparently very restrictive kind of strategy, and so we will give some leeway to player II by allowing its moves to over-spill if needed. The strategies for II we want to consider we call coding strategies. In these strategies, II only has access to player I’s latest move, and to its own most recent move. So, if $F$ is a coding strategy, and II follows it in a run of the game, then we have that for every $n$,

$T_n=F(T_{n-1},O_n)$,

with $T_{-1}=\emptyset$.

The underlying goal is to understand under which circumstances player II has a winning coding strategy in $WMG(J)$. Obviously, this is the case if II has a winning coding strategy in $RG(J)$.

Theorem 1. For an ideal $J\subset{\mathcal P}(S)$, the following are equivalent:

1. II has a winning coding strategy in $RG(J)$.
2. ${\rm cf}(\left< J\right>,{\subset})\le|J|$.

Corollary. ${\sf GCH}$ implies that for any ideal $J\subset{\mathcal P}(S)$, II has a winning strategy in $WMG(J)$.

We can reformulate our goal as asking how much one can weaken ${\sf GCH}$ in the corollary.

Let’s denote by ${\sf wSCH}$, the weak singular cardinals hypothesis, the statement that if $\kappa$ is singular strong limit of uncountable cofinality, then for no cardinal $\lambda$ of countable cofinality, we have $\kappa<\lambda<2^\kappa$.

By work of Gitik and Mitchell, we know that the negation of ${\sf wSCH}$ is equiconsistent with the existence of a $\kappa$ of Mitchell order $o(\kappa)=\kappa^{+\omega}+\omega_1$.

Theorem 2. The following are equivalent:

1. ${\sf wSCH}$.
2. For each ideal $J$ on a singular strong limit $\kappa$ of uncountable cofinality, II has a winning strategy in $RG(J)$.

We now begin the proof of Theorem 1.

$(1.\Rightarrow2.)$ Suppose II has a winning coding strategy $F$ in $RG(J)$. We want to show that ${\rm cf}(\left< J\right>,{\subset})\le|J|$. For this, we will define a map $f:J\to\left< J\right>$ with $\subset$-cofinal range, as follows: Given $X\in J$, let $T_0=X$ and $T_{n+1}=F(T_n,\emptyset)$ for all $n$. Now set

$f(X)=\bigcup_n T_n$.

To see that $f$ is cofinal, given $O\in\left< J\right>$, let $X=F(\emptyset,O)$, so that the $T_n$ are II’s responses using $F$ in a run of the game where player I first plays $O$ and then plays $\emptyset$ in all its subsequent moves. Since $F$ is winning, we must have $f(X)\supseteq O$.