Related papers: Determinacy in the Chang model
We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This…
Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…
We show that from a supercompact cardinal \kappa, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, \kappa\ and \kappa^+ are both singular, and the continuum function at \kappa\ can be…
We show that the total Cartier index of varieties with rational singularities in a bounded family is bounded. This solves a problem of Han and Jiang. The overall structure of the proof, which treats the surface case and the…
$\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a wellorderable set of cardinality less than $\Theta$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. $\mathsf{ZF + AD} + \Theta$ is regular proves that for…
We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to…
This note addresses the continuum problem, taking advantage of the breakthrough mentioned in the subtitle, and relating it to many recent advances occurring in set theory.
We present several new model-theoretic applications of the fact that, under the assumption that there exists a proper class of almost strongly compact cardinals, the powerful image of any accessible functor is accessible. In particular, we…
In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and…
Consider a linear ordering equipped with a finite sequence of monadic predicates. If the ordering contains an interval of order type \omega or -\omega, and the monadic second-order theory of the combined structure is decidable, there exists…
We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa > kappa^+$ then there…
It is shown that if there is a measurable cardinal above n Woodin cardinals and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but only iterable with respect to stacks of certain trees living between the Woodin…
We consider mainly the following version of set theory:"ZF + DC and for every $\lambda,\lambda^{\aleph_0}$ is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a…
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular…
In this paper we first formulate several ``combinatorial principles'' concerning kappa \times omega matrices of subsets of omega and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any…
We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements Filipazzi's canonical bundle formula for morphisms with connected fibres. It is then…
We give a sufficient condition for a symbolic topological dynamical system with action of a countable amenable group to be an extension of the full shift, a problem analogous to those studied by Ashley, Marcus, Johnson and others for…
For an infinite cardinal $\kappa$, let $ded\kappa$ denote the supremum of the number of Dedekind cuts in linear orders of size $\kappa$. It is known that $\kappa<ded\kappa\leq 2^{\kappa}$ for all $\kappa$ and that $ded\kappa<2^{\kappa}$ is…
Modulo the existence of large cardinals, there is a model of set theory in which for some set $B$ of regular cardinals, the sequence $\langle \text{pcf}^\alpha(B): \alpha \in \text{Ord} \rangle$ is strictly increasing. The result answers a…
V\"a\"an\"anen and Welch asked in the paper "When cardinals determine the power set: inner models and H\"artig quantifier logic" which large cardinals are consistent with the power set operation $x\mapsto P(x)$ being $\Sigma_1$-definable in…