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.


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