On the Consistency Strength of MM($ω_1$)
We prove that the consistency strength of Martin's Maximum restricted to partial orders of cardinality $ω_1$ follows from the consistency of ZFC.
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arXiv subjects
Miguel Angel Mota
Publications and source records attributed to Miguel Angel Mota.
Few new reals
We introduce a new method for building models of CH, together with $Π_2$ statements over $H(ω_2)$, 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 $ω_1$ 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.
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Corrigendum to "Measuring club-sequences together with the continuum large"
Measuring says that for e\-very sequence $(C_δ)_{δ<ω_1}$ with each $C_δ$ being a closed subset of $δ$ there is a club $C\subseteqω_1$ such that for every $δ\in C$, a tail of $C\capδ$ is either contained in or disjoint from $C_δ$. 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.
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Generalized symmetric systems and thin-very tall compact scattered spaces
We solve a well--known problem in the theory of compact scattered spaces and superatomic boolean algebras by showing that, under GCH and for each regular cardinal $κ\geq ω$, there is a poset $\mathcal P_κ$ preserving all cardinals and forcing the existence of a $κ$--thin very tall locally compact scattered space. For $κ> ω$, we conceive the poset $\mathcal P_κ$ as a higher analogue of the poset $\mathcal P_ω$ originally introduced by Asperó and Bagaria in the context of an (unpublished) alternative consistency proof.
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Forcing consequences of PFA together with the continuum large
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.
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A Generalization of Martin's Axiom
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.
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Coherent Adequate Forcing and Preserving CH
We develop a general framework for forcing with coherent adequate sets on $H(λ)$ as side conditions, where $λ\ge ω_2$ is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of $ω_2$ with finite conditions while preserving CH, solving a problem of Friedman.
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Measuring club-sequences 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 Justin Moore.
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Measuring club sequences, together with the Continuum Hypothesis
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.
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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.
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