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 study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the Δ31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Delta}^1_3}$$\end{document} level of the projective hieararchy. For Δ21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Delta}^1_2}$$\end{document} and Σ21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma}^1_2}$$\end{document} sets, the relationships between these properties follows the pattern of the well-known Cichoń diagram for cardinal characteristics of the continuum. It is known that (...) assuming suitable large cardinals, the same relationships lift to higher projective levels, but the questions become more challenging without such assumptions. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. We also prove partial results concerning Σ31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma}^1_3}$$\end{document} and Δ41\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Delta}^1_4}$$\end{document} sets. (shrink)

Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam [D. Maharam, An algebraic characterization of measure algebras, Ann. of Math. 48 154–167] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [M. Talagrand, Maharam’s problem, preprint, 31 pages, 2006] who (...) provided the first example of a Maharam algebra which is not a measure algebra. In this paper we survey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well-known properties of measure algebras. (shrink)

The main result of this paper is an improvement of the upper bound on the cardinal invariant $cov^*(L_0)$ that was discovered in [11]. Here $L_0$ is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal $non^*(L_0)$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.

It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from [3]. If (...) A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)

We show that the Filter Dichotomy Principle implies that there are exactly four classes of ideals in the set of increasing functions from the natural numbers. We thus answer two open questions on consequences of ? < ?. We show that ? < ? implies that ? = ?, and that Filter Dichotomy together with ? < ? implies ? < ?. The technical means is the investigation of groupwise dense sets, ideals, filters and ultrafilters. With related techniques we prove (...) the new inequality ?≤ cf(?). (shrink)

We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...) measure and CATEGORY = 1. Several important new metamathematical patterns among classical statements of set theory are pointed out. (shrink)

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)

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”.