Related papers: BMM is stronger than BSPFA
We study a strengthening of Bounded Martin's Maximum which asserts that if a \Sigma_1 fact holds of \omega_2^V in a stationary set preserving extension then it holds in V for a stationary set of ordinals less than \omega_2. We show that…
We prove that the consistency strength of Martin's Maximum restricted to partial orders of cardinality $\omega_1$ follows from the consistency of ZFC.
The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa…
We study a strengthening of $\mathrm{MM}^{++}$ which is called $\mathrm{MM}^{\ast, ++}$ and which was introduced by Asper\'o and Schindler. We force its bounded version $\mathrm{MM}^{\ast, ++}_{\mathfrak{c}}$, which is stronger than both…
We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\omega_1$ is measured by some club subset of…
It is known that the large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is much stronger than the Axiom of Determinacy itself. Sargsyan conjectured it to be as…
The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable…
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then it is true in all forcing extensions of V.…
We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $\Sigma_n$-$\mathsf{MP}(\mathbb R)$ and $\Pi_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle…
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…
Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and $$B_2(f):=\sup_{U}\frac{\sup_{z\in…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
The solution to fine tuning is one of the principal motivations for Beyond the Standard Model (BSM) Studies. However constraints on new physics indicate that many of these BSM models are also fine tuned (although to a much lesser extent).…
Bounded stationary reflection at a cardinal $\lambda$ is the assertion that every stationary subset of $\lambda$ reflects but there is a stationary subset of $\lambda$ that does not reflect at arbitrarily high cofinalities. We produce a…
We show that Plancherel convergence is strictly stronger than Benjamini-Schramm convergence. In order to do so, we introduce two criteria for Plancherel and Benjamini-Schramm convergence in terms of counting functions of the length…
We introduce a stronger version of an $\omega_1$-guessing model, which we call an indestructibly $\omega_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $\omega_1$-guessing models. This…
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 continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense…
We discuss the prospects for a consistent, nonlinear, partially massless (PM), gauge symmetry of bimetric gravity (BMG). Just as for single metric massive gravity, we show that consistency of BMG relies on it having a PM extension; we then…
We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $\omega_1$ can not be defined by a $\Sigma_1$ formula with parameter $A \subset \omega_1$. We show that the same conclusion holds under the…