We extend the construction of a global square sequence in extender models from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.

We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has (...) a cardinal strong up to one of its ℵ‐fixed points, and, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ. (shrink)

Assume that there is no transitive class model of [Formula: see text] with a Woodin cardinal. Let [Formula: see text] be a singular ordinal such that [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] is a regular cardinal in K. Then [Formula: see text] is a measurable cardinal in K. Moreover, if [Formula: see text], then [Formula: see text].

Given sets and a regular cardinal μ, let be the statement that for any function, there are functions and such that for all,. In, the statement is false. However, we show the theory (which is implied by + “” + “ω1 is measurable”) implies that for every there is a such that in some inner model, κ is measurable with Mitchell order.

We obtain lower bounds for the consistency strength of fully nonregular ultrafilters on ω₂.

We present a construction of a global square sequence in extender models with λ-indexing.

We prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\omega_3, \omega_2) \twoheadrightarrow (\omega_2, \omega_1)}$$\end{document} implies there is an inner model with a weak repeat measure.