Outer models and genericity
Copyright policy )Outer models and genericity
Why is forcing the only known method for constructing outer models of set theory?If V is a standard transitive model of ZFC, then a standard transitive model W of ZFC is an outer model of V if V ⊆ W and V ∩ OR = W ∩ OR.Is every outer model of a given model a generic extension? At one point Solovay conjectured that if 0# exists, then every real that does not construct 0# lies in L[G], for some G that is generic for some forcing ℙ ∈ L. Famously, this was refuted by Jensen's coding theorem. He produced a real that is generic for an L-definable class forcing property, but does not lie in any set forcing extension of L.Beller, Jensen, and Welch in Coding the universe [BJW] revived Solovay's conjecture by asking the following question: Let a ⊆ ω be such that L[a] ⊨ “0# does not exist”. Is there ab∈ L[a] such that a ∉ L[b] and a is set generic over L[b].
class-tree, Other aspects of forcing and Boolean-valued models, Set-theoretic model theory, Inner models, including constructibility, ordinal definability, and core models, outer models, class forcing, Consistency and independence results, genericity
class-tree, Other aspects of forcing and Boolean-valued models, Set-theoretic model theory, Inner models, including constructibility, ordinal definability, and core models, outer models, class forcing, Consistency and independence results, genericity
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