arXiv subjects

David Aspero

Publications and source records attributed to David Aspero.

7 recordsLinked to original sources

arXiv2022

Generalizations of Martin's Axiom, weak square, weak Chang's Conjecture, and a forcing axiom failure

We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.

math.LO↗

arXiv2021

Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $Γ$ with $λ_Γ=ω_1$, and giving rise to pairwise incompatible theories for $H_{ω_2}$.

math.LO↗

arXiv2020

Corrigendum to "Measuring club-sequences together with the continuum large"

Measuring says that for e\-very sequence $(C_\delta)_{\delta<\omega_1}$ with each $C_\delta$ being a closed subset of $\delta$ there is a club $C\subseteq\omega_1$ such that for every $\delta\in C$, a tail of $C\cap\delta$ is either contained in or disjoint from $C_\delta$. In our JSL paper "Measuring club-sequences together with the continuum large" we claimed to prove the consistency of Measuring with $2^{\aleph_0}$ being arbitrarily large, thereby answering a question of Justin Moore. The proof in that paper was flawed. In the presented corrigendum we provide a correct proof of that result. The construction works over any model of ZFC+CH and can be described as the result of performing a finite-support forcing construction with side conditions consisting of suitable symmetric systems of models with markers.

math.LO↗

arXiv2019

Parametrized Measuring and Club Guessing

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 $ω_1$ is measured by some club subset of $ω_1$. The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $σ$-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.

math.LO↗

arXiv2018

Incompatible category forcing axioms

Given a cardinal $λ$, category forcing axioms for $λ$-suitable classes $Γ$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_λ$, modulo generic extensions via forcing notions from $Γ$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of $\aleph_1$-many pairwise incompatible category forcing axioms for $ω_1$-suitable classes.

math.LO↗

arXiv2017

Few new reals

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only $\aleph_1$-many of them. Using this approach, we build a model in which a very strong form of the negation of Club Guessing at $\omega_1$ known as Measuring holds together with CH, thereby answering a well-known question of Moore. This construction can be described as a finite-support weak forcing iteration with side conditions consisting of suitable graphs of sets of models with markers. The CH-preservation is accomplished through the imposition of copying constraints on the information carried by the condition, as dictated by the edges in the graph.

math.LO↗

arXiv2011

Bounded forcing axioms and Baumgartner's conjecture

We study the spectrum of forcing notions between the iterations of $σ$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $α$-proper forcings for indecomposable countable ordinals as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of $α$-proper forcings and connect them to a hierarchy of weak club guessing principles. We show that they are, in a sense, dual to each other. In particular, these weak club guessing principles separate the bounded forcing axioms for distinct countable indecomposable ordinals. In the study of forcings completely embeddable into an iteration of $σ$-closed followed by ccc forcing, we present an equivalent characterization of this class in terms of Baumgartner's Axiom A. This resolves a well-known conjecture of Baumgartner from the 1980's.

math.LO↗