# David AsperoUniversity of East Anglia | UEA · School of Mathematics

David Aspero

Dr

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51

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Introduction

**Skills and Expertise**

## Publications

Publications (51)

Zusammenfassung
Die Arbeit [2] der gegenwärtigen Autoren amalgamierte zwei prominente Axiome der gegenwärtigen Mengenlehre, von denen vorher bekannt gewesen war, daß sie beide entscheiden, daß das Kontinuum die Größe $\aleph_2$ hat, nämlich Martins Maximum und Woodins $P_max$-Prinzip (*). Wir diskutieren dieses Resultat und seine Bedeutung für das...

We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see...

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal $\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally ext...

Measuring says that for every 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 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 $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provide...

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.

We study methods with which we can obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first prove that the consistency of a supercompact cardinal $\theta>\kappa$ implies the consistency of a forcing axiom for $\kappa$-strongly proper forcing notions which are also $\kappa$-lattice, and then eliminate the need for t...

We show that Martin's Maximum${}^{++}$ implies Woodin's ${\mathbb P}_{\rm max}$ axiom $(*)$.

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 longstanding open question from the 1970's.

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 $\omega_1$. The consistency of Strong Measuring with the negation of CH is shown, solving an open problem about parametr...

We observe 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. We also investigate some basic consequences of the Proper Forcing Axiom in ZF, and formulate a natural question about the generic absoluten...

Given a cardinal $\lambda$, category forcing axioms for $\lambda$-suitable classes $\Gamma$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_\lambda$, modulo generic extensions via forcing notions from $\Gamma$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author)....

The familiar continuum ℝ of real numbers is obtained by a wellknown procedure which, starting with the set of natural numbers ℕ = ω, produces in a canonical fashion the field of rationals ℚ and, then, the field ℝ as the completion of ℚ under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace ω by any infinite suit...

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing over a model of CH. Unlike similar constructions in the literature, our construction adds new reals, but only $\aleph_1$-many of them. Using this approach, we prove that a very strong form of the negation of Club Guessing at $\omega_...

I define a homogeneous ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}–c.c. proper product forcing for adding many clubs of ω1\documentclass...

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $\kappa$. We show the consistency of $E^{\lambda^{++},\lambda^{++}}_{\lambda\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{\lamb...

We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms.
We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large.
All these models are generic extensions via finite...

For any given uncountable cardinal κ with κ^{<κ} = κ, we present a forcing that is <κ-directed closed, has the κ^+ -cc and introduces a lightface definable well-order of H(κ+ ). We use this to define a global iteration
that adds such a well-order for all such κ simultaneously and is capable of preserving the existence of many large cardinals in the...

I construct, in ZFC, a forcing notion that collapses ℵ 3 and preserves all other cardinals. The existence of such a forcing answers a question from U. Abraham, On forcing without the con-tinuum hypothesis, J. of Symbolic Logic, vol. 48, 3 (1983), 658–661.

We isolate natural strengthenings of Bounded Martin’s Maximum which we call BMM * and A-BMM *,++ (where A is a universally Baire set of reals), and we investigate their consequences. We also show that if A-BMM *,++ holds true for every set of reals A in L(ℝ), then Woodin’s axiom (*) holds true. We conjecture that MM ++ implies A-BMM *,++ for every...

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 α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter clas...

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π2-sentences over the structure (H(ω
2), ∈, NSω1), in the sense that its (H(ω
2), ∈, NSω1) satisfies every Π2-sentence σ for which (H(ω
2), ∈, NSω1) ⊨ σ can be forced by set-forcing. In this paper we answer a question of Woodin by s...

Assuming 2ℵ0
= ℵ1 and 2ℵ1
= ℵ2, we build a partial order that forces the existence of a well-order of H(ω
2) lightface definable over ⟨H(ω
1), ∈⟩ and that preserves cardinal exponentiation and cofinalities.

We define the ℵ1.5-chain condition. The corresponding forcing axiom is a generalization of Martin’s Axiom; in fact, \({\text{MA}}_{ < \kappa }^{1.5}\) implies \({\text{M}}{{\text{A}}_{ < \kappa }}\). Also, \({\text{MA}}_{ < \kappa }^{1.5}\) implies certain uniform failures of club-guessing on ω
1 that do not seem to have been considered in the lite...

https://f1000research.com/posters/1090071

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.

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 Jus...

Measuring says that for every sequence ${\left( {{C_\delta }} \right)_{\delta with each ${C_\delta }$ being a closed subset of δ there is a club $C \subseteq {\omega _1}$ such that for every $\delta \in C$ , a tail of $C\mathop \cap \nolimits \delta$ is either contained in or disjoint from ${C_\delta }$ . We answer a question of Justin Moore by bui...

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 the Axiom A forcings. We focus on the bounded forcing axioms for the
hierarchy of $\alpha$-proper forcings and c...

There is a partial order $${\mathbb{P}}$$ preserving stationary subsets of ω
1 and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω
1 over V also collapses ω
1 over $${V^{\mathbb{P}}}$$ . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a s...

We present several forcing posets for adding a non-reflecting stationary subset of Pω1(λ), where λ≥ω2. We prove that PFA is consistent with dense non-reflection in Pω1(λ), which means that every stationary subset of Pω1(λ) contains a stationary subset which does not reflect to any set of size ℵ1. If λ is singular with countable cofinality, then den...

Several situations are presented in which there is an ordinal γ such that \({\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}\) is a stationary subset of \({[\gamma]^{\aleph_0}}\) for all stationary \({S, T\subseteq \omega_1}\). A natural strengthening of the existence of an ordinal γ for which the above conclusion...

By forcing over a model of (above ℵ0) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H(κ+) (κ≥ω2 a regular cardinal) is a well-order of H(κ+) definable over the structure 〈H(κ+),∈〉 by a parameter-free formula. Fur...

It is possible to control to a large extent, via semiproper forcing, the parameters (β0,β1) measuring the guessing density of the members of any given antichain of stationary subsets of ω1 (assuming the existence of an inaccessible limit of measurable cardinals). Here, given a pair (β0,β1) of ordinals, we will say that a stationary set S⊆ω1 has gue...

Given any subset A of ω1 there is a proper partial order which forces that the predicate x∈A and the predicate x∈ω1∖A can be expressed by -provably incompatible Σ3 formulas over the structure 〈Hω2,∈,NSω1〉. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of Hω2 definable ove...

Several results are presented concerning the existence or nonexistence, for a subset S of ω 1 , of a real r which works as a robust code for S with respect to a given sequence 〈S α :α<ω 1 〉 of pairwise disjoint stationary subsets of ω 1 , where “robustness” of r as a code may either mean that S∈L[r,〈S α * :α<ω 1 〉] whenever each S α * is equal to S...

For every uncountable regular cardinal κ, every κ-Borel partition of the space of all members of [κ] κ whose enumerating function does not have fixed points has a homogeneous club.

We prove that a form of the Erdӧs property (consistent with V = L[Hω2] and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle ψ
AC
holds, and therefore . We also prove that ψ
AC
implies that every function f: ω1 → ω1 is bounded by some canonical function on a club and us...

After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we co...

We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (ω2,ω2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω...

We isolate natural strengthenings of Bounded Martin's Maximum which we call BMM * and A–BMM *,++ (where A is a universally Baire set of reals), and we investigate their consequences. We also show that if A–BMM *,++ holds true for every set of reals A in L(R), then Woodin's axiom (*) holds true. We conjecture that MM ++ implies A–BMM *,++ for every...

This paper is mainly a survey of recent results concerning the possibility of building forcing extensions in which there is a simple definition, over the structure 〈H(ω 2 ),∈〉 and without parameters, of a prescribed member of H(ω 2 ) or of a well-order of H(ω 2 ). Some of these results are in conjunction with strong forcing axioms like PFA ++ or MM...

## Projects

Project (1)