Perfect-set forcing for uncountable cardinals

Annals of Pure and Applied Logic, 2020

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals κn : n < ω , Prikry defined the forcing P of all perfect subtrees of n<ω κn, and proved that for κ = sup n<ω κn, assuming the necessary cardinal arithmetic, the Boolean completion B of P is (ω, µ)-distributive for all µ < κ but (ω, κ, δ)distributivity fails for all δ < κ, implying failure of the (ω, κ)-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. P satisfies a Sacks-type property, implying that B is (ω, ∞, < κ)-distributive. The (h, 2)-d.l. and the (d, ∞, < κ)-d.l. fail in B. P(ω)/fin completely embeds into B. Also, B collapses κ ω to h. We further prove that if κ is a limit of countably many measurable cardinals, then B adds a minimal degree of constructibility for new ω-sequences. Some of these results generalize to cardinals κ with uncountable cofinality.

Handbook of the History of Logic, 2012

Transactions of the American Mathematical Society, 2016

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ \kappa of uncountable cofinality, while κ + \kappa ^+ enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ \kappa of uncountable cofinality where SCH fails and such that there is a collection of size less than 2 κ + 2^{\kappa ^+} of graphs on κ + \kappa ^+ such that any graph on κ + \kappa ^+ embeds into one of the graphs in the collection.

2011

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 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 $\sigma$-closed followed by ccc forcing, we present an equivalent characterization of this class in terms of Baumgartner&#x27;s Axiom A. This resolves a well-known conjecture of Baumgartner from the 1980&#x27;s.

Archive for Mathematical Logic, 2016

We introduce and study the first-order Generic Vopěnka's Principle, which states that for every definable proper class of structures C of the same type, there exist B = A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka's Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for n ∈ ω, which we introduce here, is a natural generic analogue for the C (n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka's Principle in [1]. Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, wPFA, which states that for every transitive model M in the language of set theory with some ω 1-many additional relations, if it is forced by a proper forcing P that M satisfies some Σ 1-property, then V has a transitive modelM, which satisfies the same Σ 1property, and in some set-forcing extension there is an elementary embedding fromM into M. This is a weakening of a formulation of PFA due to Schindler and Claverie [2], which asserts that the embedding fromM to M exists in V. We show that wPFA is equiconsistent with a remarkable cardinal and that wPFA implies PFA ℵ 2 , the proper forcing axiom for antichains of size at most ω 2 , but it is consistent with κ for all κ ≥ ω 2 , and therefore does not imply PFA ℵ 3 .

Annals of Pure and Applied Logic, 2001

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 ω2-Suslin tree. We also show that the axiom we call implies , as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size . Finally, we give an example of a so-called boldface bounded forcing axiom implying .

Annals of Pure and Applied Logic, 2013

We study the spectrum of forcing notions between the iterations of-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of ↵-proper forcings for indecomposable countable ordinals ↵, the Axiom A forcings and forcings completely embeddable into an iteration of a-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner's Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of ↵proper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals.

The Logica 2020 Yearbook, 2021

In this paper, I argue that one of the arguments usually put forward in defence of universism is in tension with current set theoretic practice. According to universism, there is only one set theoretic universe, V , and when applying the method of forcing we are not producing new universes, but only simulating them inside V. Since the usual interpretation of set generic forcing is used to produce a "simulation" of an extension of V from a countable set inside V itself, the above argument is credited to be a strong defence of universism. However, I claim, such an argument does not take into account current mathematical practice. Indeed, it is possible to find theorems that are available to the multiversists but that the advocate of universism cannot prove. For example, it is possible to prove results on infinite games in non-well-founded set-theories plus the axiom of determinacy (such as ZF + AFA + PD) that are not available in ZFC + P D. These results, I contend, are philosophically problematic on a strict universist approach to forcing. I suggest that the best way to avoid the difficulty is to adopt a pluralist conception of set theory and embrace a set theoretic multiverse. Consequently, the current practice of set generic forcing better supports a multiverse conception of set theory.

arXiv: Logic, 2017

Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on forcings which contain dense subsets forming topological Ramsey spaces. These forcings motivated the development of new Ramsey theory, which then was applied to the generic ultrafilters to obtain the precise structure Rudin-Keisler and Tukey orders below such ultrafilters. The content of the first two tutorials has appeared in an expository article submitted to the SEALS 2016 Proceedings. The third tutorial concentrated on uses of forcing to prove Ramsey theorems for trees which are applied to determine big Ramsey degrees of homogeneous relational structures. This is the focus of this paper.

arXiv: Logic, 2018

We introduce first the large-cardinal notion of $\Sigma_n$-super\-com\-pact\-ness as a higher-level analog of the well-known Magidor&#39;s characterization of supercompact cardinals, and show that a cardinal is $C^{(n)}$-extendible if and only if it is $\Sigma_{n+1}$-supercompact. We then develop a general framework for the preservation of $\Sigma_n$-supercompact cardinals under class forcing iterations and prove a general preservation result. As an application we obtain new proofs of the consistency of the GCH with $C^{(n)}$-extendible cardinals (Tsaprounis) and the consistency of $\rm{VP}$ with the GCH (Brooke-Taylor). Further, $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Pi_1$-definable behaviour of the power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preservi...