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The largest countable inductive set is a mouse set

Published online by Cambridge University Press:  12 March 2014

Mitch Rudominer*
Affiliation:
Department of Mathematics, Florida International University, Miami. FL 33199, U.S.A. E-mail: [email protected]

Abstract

Let κ be the least ordinal κ such that Lκ (ℝ) is admissible. Let A = {x ϵ ℝ ∣ (∃α < κ) such that x is ordinal definable in Lα (ℝ)}. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC − Replacement + “There exists ω Woodin cardinals which are cofinal in the ordinals.” T has consistency strength weaker than that of the theory ZFC + “There exists ω Woodin cardinals”, but stronger than that of the theory ZFC + “There exists n Woodin Cardinals”, for each n ϵ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = ℝ ∩ M. Since M is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every real is in A.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[Jech] Jech, Thomas, Set theory, Academic Press, Inc., San Diego, California, 1978.Google Scholar
[Kech] Kechris, A. S., The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297.Google Scholar
[KeMaSo] Kechris, A. S., Martin, D. A., and Solovay, R. M., Introduction to Q-theory, Cabal seminar 79–81, vol. 1019, Springer-Verlag, 1983, pp. 199–281.Google Scholar
[Kunen] Kunen, Kenneth, Set theory: An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North Holland, New York, N.Y., 1980.Google Scholar
[Ma1] Martin, D. A., untitled book on large cardinals and determinacy, In preparation.Google Scholar
[Ma2] Martin, D. A., The largest countable this, that, and the other, Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, 1983, pp. 97–106.Google Scholar
[MiSt] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.CrossRefGoogle Scholar
[Mo] Moschovakis, Y. N., Descriptive set theory, North-Holland, 1980.Google Scholar
[Ru] Rudominer, M., Mouse sets, Annals of Pure and Applied Logic, vol. 87 (1997), pp. 1–100.CrossRefGoogle Scholar
[St1] Steel, J. R., Scales in L(ℝ), Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, 1983, pp. 10–156.Google Scholar
[St2] Steel, J. R., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185–209.Google Scholar
[St3] Steel, J. R., Projectively wellordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104.Google Scholar
[St4] Steel, J. R., A theorem of Woodin's on mouse sets, Unpublished notes, 1996.Google Scholar