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75
On what I do not understand (and have something to say): Part I
 MATHEMATICA JAPONICA, SUBMITTED. [SH:702]; MATH.LO/9910158
, 2001
"... This is a nonstandard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve t ..."
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Cited by 40 (12 self)
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This is a nonstandard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments.
Infinite Combinatorics and Definability
, 1996
"... The topic of this paper is Borel versions of infinite combinatorial theorems. For example it is shown that there cannot be a Borel subset of [!] which is a maximal independent family. A Borel version of the delta systems lemma is proved. We prove a parameterized version of the GalvinPrikry Theorem ..."
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Cited by 24 (1 self)
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The topic of this paper is Borel versions of infinite combinatorial theorems. For example it is shown that there cannot be a Borel subset of [!] which is a maximal independent family. A Borel version of the delta systems lemma is proved. We prove a parameterized version of the GalvinPrikry Theorem. We show that it is consistent that any! 2 cover of reals by Borel sets has an! 1 subcover. We show that if V=L then there are
Normed versus topological groups: dichotomy and duality
 DISSERTATIONES MATH
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Infinite combinatorics and the foundations of regular variation
 CDAM RESEARCH REPORT SERIES
, 2008
"... The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have common c ..."
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Cited by 9 (6 self)
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The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have common combinatorial generalizations, exemplified by ‘containment up to translation of subsequences’. All of our combinatorial regularity properties are equivalent to the uniform convergence property.
The Converse Ostrowski Theorem: aspects of compactness
, 2007
"... The Ostrowski theorem in question is that an additive function bounded (above, say) on a set T of positive measure is continuous. In the converse direction, recall that a topological space T is pseudocompact if every function continuous on T is bounded. Thus theorems of ‘converse Ostrowski’ type rel ..."
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Cited by 8 (6 self)
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The Ostrowski theorem in question is that an additive function bounded (above, say) on a set T of positive measure is continuous. In the converse direction, recall that a topological space T is pseudocompact if every function continuous on T is bounded. Thus theorems of ‘converse Ostrowski’ type relate to ‘additive (pseudo)compactness’. We give a different characterization of such sets, in terms of the property of ‘generic subuniversality’, arising from the KestelmanBorweinDitor theorem and relate these to various new forms of compactness.
Forcing axioms and projective sets of reals
 Proceedings of Foundations of the Formal Sciences III, in: “Classical and new paradigms of computation and their complexity hierarchies” (Löwe
"... Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper for ..."
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Cited by 8 (0 self)
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Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + “every projective set of reals is Lebesgue measurable ” is equiconsistent with ZFC + “there is a Σ1 reflecting cardinal above a remarkable cardinal.” 1. Introduction. The current paper ∗ is in the tradition of the following result.
The classification problem for von Neumann factors
 J. Funct. Anal
"... We prove that it is not possible to classify separable von Neumann factors of types II1, II ∞ or IIIλ, 0 ≤ λ ≤ 1, up to isomorphism by a Borel measurable assignment of “countable structures ” as invariants. In particular the isomorphism relation of type II1 factors is not smooth. We also prove that ..."
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Cited by 7 (1 self)
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We prove that it is not possible to classify separable von Neumann factors of types II1, II ∞ or IIIλ, 0 ≤ λ ≤ 1, up to isomorphism by a Borel measurable assignment of “countable structures ” as invariants. In particular the isomorphism relation of type II1 factors is not smooth. We also prove that the isomorphism relation for von Neumann II1 factors is analytic, but is not Borel.
Dichotomy and infinite combinatorics: the theorems of Steinhaus
"... We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the d ..."
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Cited by 6 (3 self)
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We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski’s theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions. 1.
On Shelah's amalgamation
, 1998
"... The aim of this paper is to present a detailed explanation of three models of Shelah. We show the rule of the amalgamation in the construction of models in which all definable sets of reals have Baire property or are Lebesgue measurable. Next we construct a model in which every projective set of rea ..."
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Cited by 5 (1 self)
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The aim of this paper is to present a detailed explanation of three models of Shelah. We show the rule of the amalgamation in the construction of models in which all definable sets of reals have Baire property or are Lebesgue measurable. Next we construct a model in which every projective set of reals has Baire property, a model with the Uniformization Property and a model of ZF + DC in which all subsets of the real line are Lebesgue measurable but there is a set without Baire property.