It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated. This provides a negative (...) answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory. (shrink)

The resurrection axioms are forms of forcing axioms that were introduced recently by Hamkins and Johnstone, who developed on earlier ideas of Chalons and Veličković. In this note, we introduce a stronger form of resurrection and show that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axioms, such as Martin’s Maximum++. In addition, we study the unbounded resurrection postulates in terms of consistency lower bounds, obtaining, (...) for example, failures of the weak square principle. (shrink)

In this note, we start with the notion of a superhuge cardinal and strengthen it by requiring that the elementary embeddings witnessing this property are, in addition, sufficiently superstrong above their target. This modification leads to a new large cardinal which we call ultrahuge. Subsequently, we study the placement of ultrahugeness in the usual large cardinal hierarchy, while at the same time show that some standard techniques apply nicely in the context of ultrahuge cardinals as well.