Some questions about the lottery preparation

Question

I am trying to learn the lottery preparation technique from Hamkins' article but I am having some trouble understanding some of the technical details involved. I am at the beginning yet, so my difficulties are related with fast function forcing and the preservation of large cardinal notions when forcing with it. They are, probably, elemental difficulties arising from forcing but anyway I would like to ask here to get a deeper knowledge of this powerful technique.

For example, regarding Theorem 1.4 (preservation of weakly compactness), the author asserts that ``there are only $\kappa-$many dense sets for the forcing $\mathbb{F}_{\lambda,j(\kappa)}$'' so using enough closeness property of $M$ , and $<\kappa-closeness$ of $\mathbb{F}_{\lambda,j(\kappa)}$ (in fact, it is $\leq \lambda-$directed closed in $M$, though), we can get a $M-$generic $G$ in $V$. Why this quoted claim is true?

Also, regarding Theorem 1.5 (preservation of measurability), I find a similar trouble. Hamkins claims that ``since $2^\kappa=\kappa^+$, a simple counting argument shows that $|j(\kappa^+)|^V=\kappa^+$''. Why is this true and where it is used? I suspect that it is related to dense sets and the same argument as before applies to get a $M-$generic filter.

Every answer would be helpful. Thanks in advance!

Answer 1

I assume that $j$ is the elementary map induced by a $\kappa$-complete measure on $\kappa$. "$|j(\kappa^+)|^V=\kappa^+$" is true because every ordinal below $j(\kappa^+)$ can be represented in the ultraproduct by a function from $\kappa$ into $\kappa^+$, and the number of such functions is $(\kappa^+)^\kappa = \kappa^\kappa=2^\kappa$.

Answer 2

Notice that WLOG there exists at most $j(\kappa)-$many dense sets for your forcing. Indeed, the family $D_\alpha=\{p:\,\alpha\in dom(p)\}$ for $\alpha<j(\kappa)$ includes all the information for this forcing. For ''includes all the information'' I mean that $\bigcup \mathcal{D}=\bigcup_{\alpha<j(\kappa)} D_\alpha$ (where $\mathcal{D}$ is the family of all dense sets). Thus to get $G$ a $\mathcal{D}-$generic filter for $\mathbb{F}$ in $M$ it is enough to ensure that $G$ meets the collection of $D_\alpha$'s. With this at hand, regarding your first question, note that $j(\kappa)$ is less than $\kappa^+$ since the target model $M$ has cardinality exactly $\kappa$. For the second one, apply the same argument and Goldstern answer and you are done.