1992/11/03 by Jörg Brendle, Brendle, Jörg, Haim Judah +3 · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.math/9211202
published as Ann. Pure Appl. Logic 58 No. 3 (1992) 185--199
arxiv created 1992/11/03 · openalex publication_date 1992/11/03 · arxiv updated 2016/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 omega1V = omega1L, 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 omega1 and those on the right-hand side equal 2omega and produces a maximal almost disjoint family of subsets of omega of size omega1; 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 (2omega cap M) = 0 in N.