We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal is large. We show that for uncountable cardinals κ≤λ and , (...) if all subsets of λ either contain, or are disjoint from, a member of , then has size at least etc. As an application, we solve the Gross space problem under by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. (shrink)

We study the projective posets and their properties as forcing notions. We also define Martin's axiom restricted to projective sets, MA, and show that this axiom is weaker than full Martin's axiom by proving the consistency of ZFC + ¬lCH + MA with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists κ < 20 such that 20 < 2k”.

We strengthen a result of Harrington and Shelah by showing that, unless ω1 is an inaccessible cardinal in L, a relatively weak fragment of Martin's axiom implies that there exists a δ13 set of reals without the property of Baire.

We show that bounded forcing axioms are consistent with the existence of -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 BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...) example of a so-called boldface bounded forcing axiom implying 20=2. (shrink)

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.

Shelah, S., C. Laflamme and B. Hart, Models with second order properties V: A general principle, Annals of Pure and Applied Logic 64 169–194. We present a general framework for carrying out the construction in [2-10] and others of the same type. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the (...) universe, and the second player cheats with the aid of ♦. Section 1 contains an axiomatic framework suitable for the description of a number of related constructions, and the statement of the main theorem 1.9 in terms of this framework. In Section 2 we illustrate the use of our combinatorial principle. The proof of the main result is then carried out in Sections 3–5. (shrink)

We generalize results of [3] and [1] to hyperprojective sets of reals, viz. to more than finitely many strong cardinals being involved. We show, for example, that if every set of reals in Lω is weakly homogeneously Souslin, then there is an inner model with an inaccessible limit of strong cardinals.

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters.

In the mid-nineteenth century George Boole formulated his ‘conditions of possible experience’. These are equations and ineqaulities that the relative frequencies of events must satisfy. Some of Boole's conditions have been rediscovered in more recent years by physicists, including Bell inequalities, Clauser Horne inequalities, and many others. In this paper, the nature of Boole's conditions and their relation to propositional logic is explained, and the puzzle associated with their violation by quantum frequencies is investigated in relation to a variety of (...) approaches to the interpretation of quantum mechanics. * While preparing this paper for publication I have learnt of the untimely death of Professor J. S. Bell, and I wish to dedicate the paper to his memory. This research was undertaken while I spent a sabbatical leave at Wolfson College, and the History and Philosophy of Science Department at the University of Cambridge. I would like to thank Michael Redhead and Jeremy Butterfield for their hospitality and for helpful discussions. A first draft of this paper has been distributed among the participants of the conference 'Einstein in Context' which was held in Israel, in April 1990.1 have benefited from the comments of many colleagues. I would like to thank in particular Arthur Fine who enlightened me on the prism models, David Albert. Maya Bar-Hillel. Yemima Ben-Menachem, Mara Beller. Simon Saunders, and Mark Steiner. This research is partially supported by the Edelstein Center for the History and Philosophy of Science at the Hebrew University. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

Azriel Levy did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions (...) to set theory. (shrink)

In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions (...) and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions. (shrink)

We show that ${{\bf \Sigma}^1_3}$ -absoluteness for Sacks forcing is equivalent to the non-existence of a ${{\bf \Delta}^1_2}$ Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to ${{\bf \Sigma}^1_3}$ forcing absoluteness.

The present paper discusses a proposal which says,roughly and with several qualifications, that thecollection of mathematical truths is identical withthe set of theorems of ZFC. It is argued that thisproposal is not as easily dismissed as outright falseor philosophically incoherent as one might think. Some morals of this are drawn for the concept ofmathematical knowledge.

A strong antidiamond principle is shown to be consistent with . This principle can be stated as a “P-ideal dichotomy”: every P-ideal on ω1 either has a closed unbounded subset of ω1 locally inside of it, or else has a stationary subset of ω1 orthogonal to it. We rely on Shelah’s theory of parameterized properness for iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application (...) of the iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)