Factoring Solovay-random extensions, with application to the reduction property

Autor: Vladimir Kanovei, Vassily A. Lyubetsky
Rok vydání: 2020
Předmět:
Zdroj: Monatshefte für Mathematik. 194:105-117
ISSN: 1436-5081
0026-9255
Popis: 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 result may belong to the old set theoretic folklore. It appeared as Exapmle 1.17 in Jech’s book “Multiple forcing” without the claim that M[x] is a random extension of M in (3), but, likely, it has never been published with a detailed proof. A corollary: $${{\varvec{\Sigma }}}^{1}_{n}$$ -Reduction holds for all $$n\ge 3$$ , in models extending the constructible universe $$\mathbf{L}$$ by $$\kappa $$ -many random reals, $$\kappa $$ being any uncountable cardinal in $$\mathbf{L}$$ .
Databáze: OpenAIRE
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