# Club Guessing in the Prikry Extension

I want to use this post to construct club guessing sequences in the extension by vanilla Prikry forcing over a measurable. This was definitely known before-hand (as Todd pointed out to me), and I think this sort of thing has gotten around mostly by word of mouth. So I figured I may as well write it down since the argument is so short.

So let $\kappa$ be measurable, and let $U$ be a witnessing normal measure on $\kappa$. Let $S=(S^{\kappa^+}_\kappa)^V$, and for each $\delta\in S$ fix an increasing and continuous $f_\delta:\kappa\to\delta$ with $f_\kappa=id_\kappa$. In particular, $f_\delta$ puts $\kappa$ in bijection with a club subset of $\delta$, and so $U$ gets copied up to a normal measure $U_\delta$ over that club. Now let $\mathbb P$ be Prikry forcing over $\kappa$ with respect to $U$, and let $C$ be the generic Prikry sequence.

In $V[C]$, let $C_\delta=f_\delta ''C$ for each $\delta\in S$, and set $\bar C=\langle C_\delta : \delta\in S\rangle$. Recall that $\mathbb P$ is $\kappa^+$-cc, and so every club subset of $\kappa^+$ in the extension contains a ground model club (so in particular $S$ is stationary in $V[C]$). Next note that for every $\delta\in S$, the set $C_\delta$ is a Prikry sequence for $\delta$, and in particular will diagonalize all club subsets of $\delta$ in the ground model.

Claim: In $V[C]$, for any club $E\subseteq\kappa^+$, there is a club $D\subseteq \kappa^+$ such that, for every $\delta\in S\cap D$, we have $C_\delta\subseteq^*D$. Here $\subseteq^*$ is inclusion modulo bounded.

Proof: Let $E\subseteq\kappa^+$ be club, and let $D\subseteq E$ be a ground model club contained in $E$. Let $acc(D)$ be the set of accumulation points of $D$. If $\delta\in S\cap acc(D)$, then $D\cap\delta$ is a club subset of $\delta$ which is in the ground model, and so $C_\delta\subseteq^*D\cap\delta\subseteq E$.

Okay now (still working in the extension), let’s consider the cofinality map $cf\upharpoonright\kappa^+:\kappa^+\to\kappa$. We may as well assume that every point of $C$ is a regular cardinal. Since each $f_\delta$ was continuous, note that $cf''\bigcup_{\delta\in S}C_\delta=C$ is just our original Prikry sequence. So, we’re in the rather bizarre situation where the points that guess clubs all concentrate on a cofinal, progressive $\omega$-sequence of regular cardinals. On interesting thing to do is look at $pcf(C)$, as that would give us some insight as to how scales and club guessing sequences might interact. The following is a theorem of Jech:

Thm: Let $j:V\to M$ be a non-trivial elementary embedding where $U$ came from. Then $tcf(\prod C/J^{bd})=cf(j(\kappa))$.

So here’s a question that’s probably much easier than I’m making it: Suppose I blow up $2^\kappa=\kappa^{++}$ before singularizing $\kappa$. Is there any way in which I can make sure that I can find $j:V\to M$ with $cf(j(k))=\kappa^{++}$? If so, that’ll mean that I can guess clubs in such a way that the image of the ladder system under the cofinality map does not carry a scale. That would be kind of interesting.