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Outer models and genericity

Published online by Cambridge University Press:  12 March 2014

M. C. Stanley
M. C. Stanley*
Affiliation:
Department of Math and CS, San Jose State, San Jose, CA 95192-0103, USA, E-mail: stanley@mathcs.sjsu.edu
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Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 389 - 418
Copyright
Copyright © Association for Symbolic Logic 2003

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References

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