Haim Judah

Set Theory: On the Structure of the Real Line

This research level monograph reflects the current state of the field and provides a reference for graduate students entering the field as well as for established researchers.

The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in §1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) Cons(ZF) ⇒ Cons(ZFC + ¬ B ( m ) + ¬ U ( m ) + U ( c )). In §2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property “ F ⊆ ω ω is an unbounded family.” (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) Cons(ZF) ⇒ Cons(ZFC + U ( m ) + ¬ B ( c ) + ¬ U ( c ) + C ( c )). In §3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies “the union of the old measure zero sets is not a measure zero set,” and using this theorem we prove (ii) Cons(ZF) ⇒ Cons(ZFC + ¬ U ( m ) + C ( m ) + ¬ C ( c )).

Set theory of the continuum

Primarily consisting of talks presented at a workshop sponsored by the Mathematical Sciences Research Institute, this volume reflects a spectrum of activities in set theory. It includes a variety of research papers on the relation of set theory to algebra and topology.

The Cichoń diagram

We conclude the discussion of additivity, Baire number, uniformity, and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cichons diagram.

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.

Sets of reals

We build models where all -sets of reals are measurable and (or) have the property of Baire and (or) are Ramsey. We will show that there is no implication between any of these properties for -sets of reals.

Combinatorial properties of Hechler forcing

Brendle, J., H. Judah and S. Shelah, Combinatorial properties of Hechler forcing, Annals of Pure and Applied Logic 59 (1992) 185–199. Using a notion of rank for Hechler forcing we show: (1) assuming ωV1 = ωL1, there is no real in V[d] which is eventually different from the reals in L[ d], where d is Hechler over V; (2) adding one Hechler real makes the invariants on the left-hand side of Cichons diagram equal ω1 and those on the right-hand side equal 2ω and produces a maximal almost disjoint family of subsets of ω of size ω1; (3) there is no perfect set of random reals over V in V[ r][ d], where r is random over V and d Hechler over V[r], thus answering a question of the first and second authors.

Jumping with random reals

Abstract In the first part we study the relationship between basic properties of the ideal of measure zero sets and the properties of measure algebra. The second part is devotedto the structure of the set of random reals over models of ZFC.

Simple Cardinal Characteristics of the Continuum

We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2 and, under suitable restrictions, Pi^0_2) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) boldface Pi^0_2-characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelahs theorem that the dominating number is less than or equal to the independence number.

Iteration of Souslin forcing, projective measurability and the Borel conjecture

From an inaccessible cardinal we construct a model of ZFC where the Borel Conjecture holds and all projective sets of reals are measurable. This continues the investigation of countable support iterations of Proper Souslin forcing notions, started in a paper of Judah and Shelah.