On intermediate extensions of generic extensions by a random real

Abstract

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 $x\in M[a]$ is another real then either (1) $x\in M$, 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 ${\Sigma }_n^1$-Reduction holds for all $n\ge 3$, in a model extending the constructible universe $L$ by ${\aleph }_1$-many random reals.

11 pages

References in corpus (3)

• The Bristol Model: an abyss called a Cohen real
• Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
• On Harrington's model in which Separation holds but Reduction fails at the 3rd projective level, and on some related models of Sami