Related papers: Sigma-Prikry forcing I: The Axioms
We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete $\Pi^1_1$ set…
Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to $\kappa$) such that the composition still…
We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $\Sigma_1^1$-complete quasi-order in the generalized Borel-reducibility hierarchy ($\kappa>\omega$). This improvement to known results in $L$ has many new…
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 study the spectrum of forcing notions between the iterations of $\sigma$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals as well as…
In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(\omega_3,\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paper we formulate a…
We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of…
This paper is meant to present in a coherent way several instances of quite common phenomena that was first identified (independently) by Bukovsk\'y and Dehornoy. We present the basic result for Prikry type forcing and show how to extend it…
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…
Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square…
In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset…
We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not…
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…
We establish the consistency of the failure of the diamond principle on a cardinal $\kappa$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a…
We show that in a category with pullbacks, arbitrary sifted colimits may be constructed as filtered colimits of reflexive coequalizers. This implies that "lex sifted colimits", in the sense of Garner--Lack, decompose as Barr-exactness plus…
We show that the forcing axiom for countably compact, $\omega_2$-Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah and sets a new limit to the generalization of Martin's Axiom to the stage…
Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $\delta$ of countable cofinality and every regular uncountable $\mu<\delta$ there are stationarily many non-approachable…
It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if…
In this paper we continue the study in [Gilton-Levine-Stejskalova] of compactness and incompactness principles at double successors, focusing here on the case of double successors of singulars of countable cofinality. We obtain models which…
We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.