Arnold W. Miller

Special Subsets of the Real Line

Publisher Summary This chapter discusses some peculiar sets of real numbers and some of the methods for obtaining them. Bernstein constructed a set of reals of cardinality the continuum, which is neither disjoint from nor contains an uncountable closed set. His construction used transfinite induction and the fact that every uncountable closed set has cardinality the continuum. A set of reals is meager if it is the countable union of nowhere dense sets. A set of reals is comeager if it is the complement of a meager set. The Baire category theorem says that no complete metric space is meager in itself. Assuming the continuum hypothesis, there is a set of reals of cardinality the continuum that has countable intersection with every measure zero set. A set of reals X has universal measure zero if for all measures μ on the Borel sets, there is a Borel set of μ-measure zero covering X. The existence of uncountable sets of universal measure zero and uncountable perfectly meager sets does not require any axioms beyond the usual Zermelo–Fraenkel with the axiom of choice. There exists a set of reals X of cardinality ω 1 which has universal measure zero and is perfectly meager.

γ-sets and other singular sets of real numbers

Abstract A family of J of open subsets of the real line is called an ω-cover of a set X iff every finite subset of X is contained in an element of J . A set of reals X is a γ-set iff for every ω-cover J of X there exists 〈D n : n J ω such that X⊆ ∪ n ∩ m > n D m . In this paper we show that assuming Martins axiom there is a γ-set X of cardinality the continuum.

Infinite combinatorics and definability

Abstract 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 [ω] ω which is a maximal independent family. A Borel version of the delta systems lemma is proved. We prove a parameterized version of the Galvin-Prikry Theorem. We show that it is consistent that any ω 2 cover of reals by Borel sets has an ω 1 subcover. We show that if V \= L, then there are π 1 1 Hamel bases, maximal almost disjoint families, and maximal independent families.

Descriptive set theory over hyperfinite sets

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

The γ-borel conjecture

Abstract.In this paper we prove that it is consistent that every γ-set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong γ-set is countable while not every γ-set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.

Sacks forcing, Laver forcing, and Martin's axiom

SummaryIn this paper we study the question assuming MA+⌝CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewskis ideals0. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.

On Q-sets

This is the published version, also available here: http://www.dx.doi.org/10.1090/S0002-9939-1980-0550513-4. First published in Proc. AMS. in 1980, published by the American Mathematical Society.

Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way

These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Part 1 is concerned with Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies and MA. Section 14 and 15 contain new results concerning the lengths of Borel hierarchies in the Cohen and random real model. Part 2 contains standard results on the theory of Analytic sets. Section 25 contains Harringtons Theorem that it is consistent to have

Point-cofinite covers in the Laver model

\Pi^1_2

Selective covering properties of product spaces

sets of arbitrary cardinality. Part 3 has the usual separation theorems. Part 4 gives some applications of Gandy forcing. We reverse the usual trend and use forcing arguments instead of Baire category. In particular, Louveaus Theorem on