arXiv subjects
Publications and source records attributed to David Asperó.
We force the existence of a chain of length $ω_3$ in $[ω_1]^{ω_1}$ increasing modulo finite. The construction involves symmetric systems of models of two types as side conditions, introduced by the second author. This improves previous results of Koszmider and Veličković-Venturi.
We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $κ\geqω_2$ is a regular cardinal, we prove that there is a proper and $\aleph_2$-c.c.\ forcing giving rise to a model of this forcing axiom together with $2^{\aleph_0}=κ$ and which, in addition, satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \exists yφ(a, y)$, where $a\in \mathcal{H}(\aleph_2)$ and $φ(x, y)$ is a $Σ_0$ formula with the property that for every ground model $M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice poset -- specifically, a poset $\mathbb{Q}\subseteq\mathcal{H}(κ)^M$ which is $ω_1$-linked and symmetrically proper -- adding some $b$ such that $φ(a, b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for $\aleph_1$-generated ideals on $ω_1$, among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin's Maximum for posets of size $\aleph_1$.
Starting from the existence of a weakly compact cardinal, we build a generic extension of the universe in which $GCH$ holds and all $\aleph_2$-Aronszajn trees are special and hence there are no $\aleph_2$-Souslin trees. This result answers a well-known open question from the 1970's.
We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $θ>κ$ to get the consistency of the forcing axiom for $κ$-strongly proper forcing notions which are also $κ$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $κ$-strongly proper forcings. We also produce a model of this forcing axiom with $2^κ$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom.
We show that Martin's Maximum${}^{++}$ implies Woodin's ${\mathbb P}_{\rm max}$ axiom $(*)$. This answers a question from the 1990's and amalgamates two prominent axioms of set theory which were both known to imply that there are $\aleph_2$ many real numbers.
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to DC-preserving symmetric submodels of forcing extensions. Hence, ZF+DC not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF+DC and ZFC. Our results confirm ZF+DC as a natural foundation for a significant portion of "classical mathematics" and provide support to the idea of this theory being also a natural foundation for a large part of set theory.
Methods of Higher Forcing Axioms was a small workshop in Norwich, taking place between 10--12 of September, 2019. The goal was to encourage future collaborations, and create more focused threads of research on the topic of higher forcing axioms. This is an improved version of the notes taken during the meeting by Asaf Karagila.
Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $κ$. We show the consistency of $E^{λ^{++},λ^{++}}_{λ\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_λ$ in the space $(λ^{++})^{λ^{++}}$, being continuously reducible to $E^{2,λ^{++}}_{λ^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_{λ^+}$ in the space $2^{λ^{++}}$. Then we show the consistency of $E^{2,κ}_{reg}$, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space $2^κ$, being $Σ_1^1$-complete. We finish by showing, for $Π_2^1$-indescribable $κ$, that the isomorphism relation between dense linear orders of cardinality $κ$ is $Σ_1^1$-complete.
We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.
We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $ω_1$ that don't seem to have been considered in the literature before.
One of the most frustrating problems faced by set theorists working with iterated proper forcing is the lack of techniques for producing models in which the continuum has size greater than the second uncountable cardinal. In this paper we solve this problem in the specific case of measuring, a very strong negation of Club Guessing introduced by Justin Moore.
We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and with symmetry constraints. Also, we show that a small variation of this construction produces a model of measuring together with the continuum being larger than the second uncountable cardinal.