2018/11/26 by Vladimir Kanovei, Kanovei, Vladimir, Vassily Lyubetsky +1
Computer Science · Mathematics · #03E15 #03E35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1811.10568
openalex publication_date 2018/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real a is random over a model M and x∈ M[a] is another real then either (1) x∈ M, or (2) M[x]=M[a], or (3) M[x] is a random extension of M and M[a] is a random extension of M[x]. This is a less-known result of old set theoretic folklore, and, as far as we know, has never been published. As a corollary, we prove that Σ1n-Reduction holds for all n≥3, in a model extending the constructible universe L by ℵ1-many random reals.