arXiv subjects
Publications and source records attributed to Haim Judah.
We show that (1) If ZF is consistent then the following theory is consistent "ZF + DC(omega_{1}) + Every set of reals has Baire property" and (2) If ZF is consistent then the following theory is consistent "ZFC + `every projective set of reals has Baire property' + `any union of omega_{1} meager sets is meager' ".
We describe a method of building ``nice'' sigma-ideals from Souslin ccc forcing notions. [These notes were written down in 1992, but were not submitted to any journal. In a slightly modified form, they were incorporated to: T. Bartoszynski and H. Judah, Set Theory: on the structure of the real line, A K Peters, Wellesley, MA, 1995; pages 193-203.]
We study the ideal of meager sets and related ideals.
The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.
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 Cichon's diagram.
We will kill the old Luzin and Sierpinski sets in order to build a model where U(Meager) = U(Null)= aleph_1 and there are neither Luzin nor Sierpinski sets. Thus we answer a question of J. Steprans, communicated by S. Todorcevic on route from Evans to MSRI.
We give a model where there is a ccc Souslin forcing which does not satisfy the Knaster condition. Next, we present a model where there is a sigma-linked not sigma-centered Souslin forcing such that all its small subsets are sigma-centered but Martin Axiom fails for this order. Furthermore, we construct a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which is proper but not ccc that does not contain a perfect set of mutually incompatible conditions. Finally we show that ccc Sigma^1_2-notions of forcing may not be indestructible ccc.
If ZFC is consistent, then each of the following are consistent with ZFC + 2^{aleph_0}= aleph_2 : 1.) X subseteq R is of strong measure zero iff |X| <= aleph_1 + there is a generalized Sierpinski set. 2.) The union of aleph_1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size aleph_2.
We show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property.
In this work we use a notion of rank first introduced by James Baumgartner and Peter Dordal and later developed independently by the third author to show that adding a Hechler real has strong combinatorial consequences. We prove: 1) assuming omega_1^V = omega_1^L, 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 Cicho'n's diagram equal omega_1 and those on the right-hand side equal 2^omega and produces a maximal almost disjoint family of subsets of omega of size omega_1; 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. As an intermediate step in the proof of 3) we show that given models M subseteq N of ZFC such that there is a perfect set of random reals in N over M, either there is a dominating real in N over M or mu (2^omega cap M) = 0 in N.
We discuss the relationship between perfect sets of random reals, dominating reals, and the product of two copies of the random algebra B. Recall that B is the algebra of Borel sets of 2^omega modulo the null sets. Also given two models M subseteq N of ZFC, we say that g in omega^omega cap N is a dominating real over M iff forall f in omega^omega cap M there is m in omega such that forall n geq m (g(n) > f(n)); and r in 2^omega cap N is random over M iff r avoids all Borel null sets coded in M iff r is determined by some filter which is B-generic over M. We show that there is a ccc partial order P which adds a perfect set of random reals without adding a dominating real, thus answering a question asked by the second author in joint work with T. Bartoszynski and S. Shelah some time ago. The method of the proof of this result yields also that B times B does not add a dominating real. By a different argument we show that B times B does not add a perfect set of random reals (this answers a question that A. Miller asked during the logic year at MSRI).