Related papers: Stationary Reflection and the failure of SCH

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located…

The physical properties of bound-state charged massive scalar field configurations linearly coupled to a spherically symmetric charged reflecting shell are studied {\it analytically}. To that end, we solve the Klein-Gordon wave equation for…

It is shown that the consistency strength of ZF + DC + "the closed unbounded ultrafilter on omega_1 is an ultrafilter" is exactly ZFC + one measurable cardinal.

Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $\omega_1$ is preserved by any proper forcing. We…

We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals.

For a light scalar coupled to gravity, I study the gravitational backreaction associated with large field variations. I show a generic obstruction in sourcing a super-Planckian scalar profile without making the whole experiment collapse…

A usual dichotomy is that in many cases, reasonably definable sets, satisfy the CH, i.e. if they are uncountable they have cardinality continuum. A strong dichotomy is when: if the cardinality is infinite it is continuum as in [Sh:273]. We…

We prove that the upper bounds for the consistency strength of certain instances of mutual stationarity considered by Liu-Shelah~\cite{MR1469093} are close to optimal. We also consider some related and, as it turns out, stronger properties.

Covering matrices were introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of his work and in subsequent work with Sharon, he isolated two reflection principles,…

This is a revised version (of late 2020) of [Sh:700], which is arXiv:math/0012170 . First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency $ \mathfrak{b} = \mathfrak{d} =…

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a…

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^{\mu} > \mu^+$ then $\binom{\mu^+}{\mu}\to \binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every ordinal $\tau<\mu^+$. We obtain an optimal positive relation under $2^\mu = \mu^+$,…

We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to…

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods…

In this paper we investigate more characterizations and applications of $\delta$-strongly compact cardinals. We show that, for a cardinal $\kappa$ the following are equivalent: (1) $\kappa$ is $\delta$-strongly compact, (2) For every…

We prove the following consistency result for cardinal sequences of length $< \om_3$: if GCH holds and $\la \geq \om_2$ is a regular cardinal, then in some cardinal-preserving generic extension $2^{\om} = \la$ and for every ordinal $\eta <…

We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let $M=(V^M,P)$ be a countable model of second order set theory $\mathsf{ZF}_2$ (with universe $V^M$ and classes $P$) which models "$\kappa$…

We show that the consistency strength of the system NFUB, a variant of Quine's "New Foundations" recently introduced by Randall Holmes, is precisely that of [ZFC - Power Set] + "There is a weakly compact cardinal''. This is a preliminary…

We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S…

We answer a question of Krueger by obtaining -- from countably many Mahlo cardinals -- a model where there is a disjoint stationary sequence on $\aleph_{n+2}$ for every $n\in\omega$. In that same model, the notions of being internally…