Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-14T00:28:55.690Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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