Veličković's model game

Question

This question is about the argument for Lemma 3.7 in Forcing axioms and stationary sets (MSN) by Boban Veličković.

He defines a game $G_\alpha$ between two players, playing objects in $H_\kappa$, depending on a function $F : [\kappa]^{<\omega} \to \kappa$. At stage $n$, Player I plays an interval of ordinals $I_n$ below $\kappa$ and an ordinal $\xi_n \in I_n$. Player II plays an ordinal $\mu_n$. $\min(I_{n+1})$ must be above $\mu_n$. Player I wins if the closure $M$ of $\{ \xi_n : n < \omega \} \cup \alpha$ under $F$ is contained in $\bigcup_n I_n$, and $M \cap \omega_1 = \alpha$.

Now I understand the argument for why player I has a winning strategy when we ignore the requirement on $\alpha$. For the $\alpha$-indexed games, he says: Let $A_F$ be the set of $\alpha<\omega_1$ such that player II has a winning strategy $\sigma_\alpha$. "Since $A_F$ has cardinality $\leq \aleph_1$, there is a strategy $\sigma$ which dominates all the $\sigma_\alpha$. It follows that $\sigma$ is a winning strategy for II in $G_\alpha$ for every $\alpha \in A_F$."

What does it mean to say that a strategy "dominates" a lot of other ones (for the same player), and why does such $\sigma$ exist?

Answer 1

What constitutes a partial play doesn't depend on $\alpha$. And $\kappa$ is assumed to have cofinality strictly larger than $\aleph_1$. So he means: define $\sigma$ applied to a partial play to just be some ordinal below $\kappa$ that is above all the outputs of the $\aleph_1$-many strategies $\sigma_\alpha$.

(And it is easy to see that for any fixed $\alpha$, if $\sigma_\alpha$ is a winning strategy for II in the game $G_\alpha$, then so is any function from partial runs into $\kappa$ that is $\ge \sigma_\alpha$.)