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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

REFERENCES

[B1]Barwise, J., Infinitary logic and admissible sets, Ph.D. thesis, Stanford University, 1967.Google Scholar
[B2]Barwise, J., Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, New York, 1975.CrossRefGoogle Scholar
[BJW]Beller, A., Jensen, R. B., and Welch, P., Coding and Universe, London Mathematical Society Lecture Note Series, vol. 47, Cambridge University Press, 1982.CrossRefGoogle Scholar
[C]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, 1966.Google Scholar
[HF]Friedman, H., Countable models of set theories, Cambridge summer school in logic, Springer-Verlag, 1973.Google Scholar
[F1]Friedman, Sy, The genericity conjecture, this Journal, vol. 59 (1994), pp. 606–614.Google Scholar
[F2]Friedman, Sy, Fine structure and class forcing, de Gruyter, 2000.CrossRefGoogle Scholar
[Kc]Kechris, A. S., Classical descriptive set theory, Springer-Verlag, 1994.Google Scholar
[Ks]Keisler, H. J., Forcing and the omitting types theorem, Studies in model theory (Morley, M. D., editor), vol. 8, Mathematics Assosiation of America, 1973, pp. 96–133.Google Scholar
[Sh]Shoenfield, J. R., Unramified forcing, Proceedings of the symposium for pure mathematics, American Mathematical Society, 1971.Google Scholar
[S1]Stanley, M. C., A non-generic real incompatible with 0#, Annals of Pure and Applied Logic, vol. 85 (1997), pp. 157–192.CrossRefGoogle Scholar
[S2]Stanley, M. C., Invisible genericity and 0#, this Journal, vol. 63 (1998), pp. 1297–1318.Google Scholar
[W]Woodin, W. H., The axiom of determinacy, forcing axioms, and the non-stationary ideal, de Gruyter, 2000.Google Scholar