Note on covering and approximation properties (Infinitary combinatorics in set theory and its applications)
拓史 酒井 · Kyoto University Research Information Repository (Kyoto University) · 2015
We discuss the covering and approximation properties of an ultrapower of $V$ by a $\kappa$ -complete ultrafilter over a measurable cardinal $\kappa$ .Among other things, we prove that it can have both of the $\mu$ -covering and $\mu$-approximation properties for every cardinal $\mu>\kappa^{+}.$ 1 IntroductionIn this paper we discuss the covering and approximation properties of inner models, which were introduced by Hamkins [3].First recall these properties: Let $M$ be an inner model, i,e. a transitive inner model of ZFC containing all ordinals, and let $\mu$ be a cardinal (in $V$ ).Note that $|x|<\mu$ if and only if $|x|^{M}<\mu$ for any set $x\in M$ .We say that $M$ has theThese properties are often discussed in the context of forcing extensions.It was proved in [3] that if $V$ is a set forcing extension of $M$ by a poset $\mathbb{P}$ , and $\mu$ is a cardinal with $|\mathbb{P}|^{M}<\mu$ , then $M$ has the $\mu$-covering and $\mu$ -approximation properties.Using this fact, Laver [4] proved that a ground model is definable in any set forcing extensions.It was also used in [3] and [4] to prove that certain large cardinals are not created by small forcing extensions.Similar use of these properties can be found in Reiz [5], Fuchs-Hamkins-Reiz [2] and Viale-WeiB [6], too.In this paper we study the covering and approximation properties of an ultrapower of $V$ by a $\kappa$ -complete ultrafilter over a measurable cardinal $\kappa$ .Throughout this paper let $\kappa,$ $U,$ $M$ and $j$ be as follows:Moreover, for each $f\in\kappa V$ , let $(f)_{U}\in\kappa V/U$ be the equivalence class represented by $f,$ and let $[f]_{U}\in M$ be the target of $(f)_{U}$ by the transitive collapse o $f^{\kappa}V/U.$Here we summarize our results in this paper.First we present those on the convering property.Note that $M$ has the $\mu$ -covering property for every cardinal $\mu\leq\kappa^{+}$ because $\kappa M\subseteq M$ .We will obtain the following: