Bounded forcing axioms and the continuum

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 Journal of Symbolic Logic

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma $ : the hierarchy of bounded forcing axioms, of $\Sigma ^1_1$ -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level $\kappa $ says that whenever T is a tree of height $\omega _1$ and width $\kappa $ that does not have a branch of order type $\omega _1$ , and whenever ${\mathord {\mathbb P}}$ is a forcing notion in $\Gamma $ , then it is not the case that ${\mathord {\mathbb P}}$ forces that T has such a branch. $\Sigma ^1_1$ -absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don’t add reals, the three principles at level $2^\omega $ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don’t add reals.

We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω 1 ) which is ∆ 1 definable with parameter a subset of ω 1 . Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes ℵ 2 and also satisfies BPFA must contain all subsets of ω 1 . We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the Härtig quantifier is not lightface projective.

Lecture Notes in Computer Science, 2013

We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory.

arXiv (Cornell University), 2019

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.

Journal of Mathematical Logic

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 text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. 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 [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].

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 .

1998

We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the solutions we develop various ideas interesting per se.These include a new iterable condition for ``not adding Cohen reals'' (which has a flavour of preserving special properties of p-points), new intriguing properties of ultrafilters (weaker than being Ramsey but stronger than p-point) and some new applications of variants of the PP--property.

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 2008

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 P F A ++ or M M , some are not. I also observe (Corollary 4.4) that the existence of certain objects of size ℵ 1 follows outright from the existence of large cardinals. This observation is motivated by an attempt to extend the P F A ++ result to a result mentioning M M ++. 1 Main starting questions and some pieces of notation The work presented here deals mostly with the problem of finding optimal definitions of well-orders of the reals and other objects. More precisely, it addresses the following two questions. 1 Question 1: Suppose A is a subset of ω 1. Suppose we are given the task of going over to a nice set-forcing extension 2 in which A admits a simple definition Φ(x), without parameters, over the structure H(ω 2), ∈ or over some natural (definable) extension of this structure, like H(ω 2), ∈, N S ω 1 .

Annals of Mathematical Logic, 1980