Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T18:36:32.935Z Has data issue: false hasContentIssue false
  • English
  • Français

Proper forcing and L(ℝ)

Published online by Cambridge University Press:  12 March 2014

Itay Neeman  and
Jindřich Zapletal
Itay Neeman*
Affiliation:
Harvard University, Department of Mathematics, One Oxford St., Cambridge, MA 02138, USA
Jindřich Zapletal*
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA
*
Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA, E-mail: [email protected]
Department of Mathematics, University of Florida, Gainsville, FL 32611-8105, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L(ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of ADL(ℝ): roughly ω Woodin cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 801 - 810
Copyright
Copyright © Association for Symbolic Logic 2001

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

[1]Foreman, Matthew and Magidor, Menachem, Large cardinals and definable counter examples to the continuum hypothesis, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 47–97.CrossRefGoogle Scholar
[2]Hauser, Kai, Mathias, Adrian D. R., and Woodin, W. Hugh, The axiom of determinacy. (Forthcoming).Google Scholar
[3]Jech, Thomas, Set theory, Academic Press, 1978.Google Scholar
[4]Jech, Thomas, Multiple forcing, Cambridge University Press, 1987.CrossRefGoogle Scholar
[5]Martin, Donald A. and Steel, John, Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 1–73.CrossRefGoogle Scholar
[6]Neeman, Itay and Zapletal, Jindřich, Proper forcing and L(ℝ), an eprint version of this paper, with an added Appendix. It can be obtained from the URL address http://wuw.arXiv.org/abs/math/0003027.Google Scholar
[7]Neeman, Itay, Proper forcing and absoluteness in L(ℝ), Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 281–301.Google Scholar
[8]Steel, John, Inner models with many Woodin cardinals, Annals of pure and applied logic, vol. 65 (1993), no. 2, pp. 185–209.CrossRefGoogle Scholar
[9]Woodin, W. Hugh, The axiom of determinacy, forcing axioms, and the non-stationary ideal, de Gruyter, 1999.CrossRefGoogle Scholar