The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the...

In the author's 2012 paper, the V-definable Stable Core 𝕊 = (L[S],S) was introduced. It was shown that V is generic over 𝕊 (for 𝕊-definable dense classes), each V-definable club contains an 𝕊-definable club, and the same holds with 𝕊 replaced by (HOD,S), where HOD denotes Gödel's inner model of hereditarily ordinal-definable sets. In the present article we extend this to models of class theory by introducing the V-definable Enriched Stable Core 𝕊* = (L[S*],S*). As an application we obtain...

Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $⟨S_{\alpha }:\alpha <\omega ₁⟩$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S\in L[r,⟨S{*}_{\alpha }:\alpha <\omega ₁⟩]$ whenever each $S{*}_{\alpha }$ is equal to $S_{\alpha }$ modulo nonstationary changes, or may have the weaker meaning that $S\in L[r,⟨S_{\alpha }\cap C:\alpha <\omega ₁⟩]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that S gets exactly...

In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length $\omega$ is sometimes the smallest common extension of this sequence and very often it is not.

We present a reformulation of the fine structure theory from Jensen [72] based on his Σ* theory for K and introduce the Fine Structure Principle, which captures its essential content. We use this theory to prove the Square and Fine Scale Principles, and to construct Morasses.

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that consistently there exists an uncountable coanalytic subset of the plane that intersects every C¹ curve in a countable set.