On -Strongly Measurable Cardinals

Abstract

We prove several consistency results concerning the notion of $\omega $ -strongly measurable cardinal in $\operatorname {\mathrm {HOD}}$ . In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa ) = \kappa $ , that every successor of a regular cardinal is $\omega $ -strongly measurable in $\operatorname {\mathrm {HOD}}$ .

On $Σ_1$-Definable Closed Unbounded Sets

“…Recent developments in the study of canonical inner models of set theory provide strong motivations for analyzing the definability, specifically, the ordinal definability, of these objects. In particular, the notion of a ω-strongly measurable cardinal κ in HOD, introduced by Woodin (see [19,Definition 189]) to measure local failures of the inner model HOD to approximate the set-theoretic universe V , is equivalent to the fact that the restriction of the closed unbounded filter on {α < κ | cof(α) = ω} to HOD identifies with the intersection of a small number of ordinal definable normal measures on κ in HOD (see [2,Lemma 2.4]). Since, in HOD, many of the sets in this intersection do not contain a closed unbounded subset, it follows that there are many subsets of the cardinal κ that are HOD-stationary (i.e., that meet every closed unbounded subset of κ that is an element of HOD), but are not stationary subsets of κ in V .…”

“…1 See [8, p. 183]. 2 More precisely, it is possible to use [13,Lemma 2.3] to show that, if ϕ(v 0 , . .…”