1992/09/15 by Jörg Brendle, Brendle, Jörg, Haim Judah +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rough Sets and Fuzzy Logic #math.LO
paper · pdf · doi:10.48550/arxiv.math/9209205
arxiv created 1992/09/15 · openalex publication_date 1992/09/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 2omega modulo the null sets. Also given two models M subseteq N of ZFC, we say that g in omegaomega cap N is a dominating real over M iff forall f in omegaomega cap M there is m in omega such that forall n geq m (g(n) > f(n)); and r in 2omega 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).