Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-08T08:07:30.819Z Has data issue: false hasContentIssue false
  • English
  • Français

HODL(ℝ) is a Core Model Below Θ

Published online by Cambridge University Press:  15 January 2014

John R. Steel
John R. Steel*
Affiliation:
Department of Mathematics, University of California, Los Angeles, Ca 90024.E-mail: steel@math.ucla.edu
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we shall answer some questions in the set theory of L(ℝ), the universe of all sets constructible from the reals. In order to do so, we shall assume ADL(ℝ), the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L(ℝ) are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L(ℝ), and for reasons we cannot discuss here, ZFC + ADL(ℝ) yields the most interesting “completion” of the ZFC-theory of L(ℝ).

ADL(ℝ) implies that L(ℝ) satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L(ℝ). Nevertheless, there is a natural inner model of L(ℝ), namely HODL(ℝ), which satisfies ZFC. (HOD is the class of all hereditarily ordinal definable sets, that is, the class of all sets x such that every member of the transitive closure of x is definable over the universe from ordinal parameters (i.e., “OD”). The superscript “L(ℝ)” indicates, here and below, that the notion in question is to be interpreted in L(R).) HODL(ℝ) is reasonably close to the full L(ℝ), in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL(ℝ): what is its first order theory, and in particular, does it satisfy GCH?

These questions first drew attention in the 70's and early 80's. (See [4, p. 223]; also [12, p. 573] for variants involving finer notions of definability.)

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 1 , Issue 1 , March 1995 , pp. 75 - 84
Copyright
Copyright © Association for Symbolic Logic 1995

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] Becker, H., Thin collections of sets of projective ordinals and analogs of L, Annals of Mathematical Logic, vol. 19 (1980), pp. 205–241.CrossRefGoogle Scholar
[2] Becker, H. and Kechris, A. S., Sets of ordinals constructible from trees and the third Victoria Delfino problem, Contemporary mathematics, vol. 31 (1984), pp. 13–29.CrossRefGoogle Scholar
[3] Becker, H. and Moschovakis, Y. N., Measurable cardinals in playful models, Cabal seminar 77–79 (Kechris, A. S. et al., editors), Springer Lecture Notes in Mathematics, vol. 839, Springer-Verlag, New York, 1981, pp. 203–215.CrossRefGoogle Scholar
[4] Delfino, V., Victoria Delfino problem list, Cabal seminar 81–85 (Kechris, A. S. et al., editors), Springer Lecture Notes in Mathematics, no. 1333, Springer-Verlag, New York, 1988, p. 221.Google Scholar
[5] Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[6] Jensen, R. B., Inner models and large cardinals, forthcoming in This Journal.Google Scholar
[7] Kechris, A. S., AD and projective ordinals, Cabal seminar 76–77 (Kechris, A.S. et al., editors), Springer Lecture Notes in Mathematics, vol. 689, Springer-Verlag, 1978, pp. 91–132.CrossRefGoogle Scholar
[8] Martin, D. A. and Steel, J. R., The extent of scales in L (ℝ), Cabal seminar 79–81 (Kechris, A. S. et al., editors), Springer Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, New York, 1983, pp. 86–96.CrossRefGoogle Scholar
[9] Martin, D. A. and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar
[10] Martin, D. A. and Steel, J. R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73.CrossRefGoogle Scholar
[11] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Springer Lecture Notes in Logic, vol. 3, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
[12] Moschovakis, Y. N., Descriptive set theory, North Holland, Amsterdam, 1980.Google Scholar
[13] Steel, J. R., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185–209.CrossRefGoogle Scholar
[14] Woodin, H., Omega Woodin cardinals from AD, unpublished lecture notes taken by E. Schimmerling.Google Scholar
[15] Woodin, H., Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Science of the USA, vol. 85, pp. 6587–6591.CrossRefGoogle Scholar