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Some applications of coarse inner model theory

Published online by Cambridge University Press:  12 March 2014

Greg Hjorth
Greg Hjorth*
Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, Ca 90095-1555, USA, E-mail: [email protected]
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Abstract

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .

Keywords

Inner model theorylarge cardinalsequivalence relations
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 337 - 365
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Dodd, A., The core model, London Mathematical Society Lecture Note Series, vol. 61, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
[2]Foreman, M. and Magidor, M., Large cardinals and definable counterexamples to the continuum hypothesis, to appear in the Annals of Pure and Applied Logic.Google Scholar
[3]Friedman, S., Analytic equivalence relations and illfounded models, preprint.Google Scholar
[4]Harrington, L., Long projective wellorderings, Annals of Mathematical Logic, vol. 12 (1977), pp. 1–24.CrossRefGoogle Scholar
[5]Harrington, L. and Kechris, A., On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1991), pp. 109–154.CrossRefGoogle Scholar
[6]Harrington, L. and Sami, R., Equivalence relations, projective and beyond, North-Holland, Amsterdam.CrossRefGoogle Scholar
[7]Harrington, L. and Shelah, S., Counting equivalence classes for co-k-Souslin equivalence relations, North-Holland, Amsterdam, 1982.CrossRefGoogle Scholar
[8]Harrrington, L., Analytic determinacy and 0#, this Journal, vol. 43 (1978), pp. 685–693.Google Scholar
[9]Hauser, K., The consistency strength of projective absoluteness, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 245–295.CrossRefGoogle Scholar
[10]Hjorth, G., Thin equivalence relations and effective decompositions, this Journal, vol. 58 (1993), pp. 1153–1164.Google Scholar
[11]Hjorth, G., The size of the ordinal μ2, Journal of the London Mathematical Society, vol. 52 (1995), pp. 417–433.CrossRefGoogle Scholar
[12]Hjorth, G., Two applications of inner model theory in the study of sets, Bulletin of Symbolic Logic, vol. 2 (1996), pp. 94–107.CrossRefGoogle Scholar
[13]Hjorth, G., Variations on the Martin-Solovay tree, this Journal, vol. 61 (1996), pp. 40–51.Google Scholar
[14]Jackson, S., Partition properties and well-ordered sequences, Annals of Pure and Applied Logic, vol. 48 (1990), pp. 81–101.CrossRefGoogle Scholar
[15]Jech, J., Set theory, Academic Press, New York, 1978.Google Scholar
[16]Kechris, A., On transfinite sequences of projective sets with an application to equivalence relations, North Holland, Amsterdam, 1978.CrossRefGoogle Scholar
[17]Kechris, A., Martin, D., and Solovay, R., Introduction to q-theory, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, Berlin, 1983.Google Scholar
[18]Martin, D. and Steel, J., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–74.CrossRefGoogle Scholar
[19]Mitchell, W. and Steel, J., The fine structure of iteration trees, Lecture Notes in Logic Series, Springer-Verlag, Berlin, 1995.Google Scholar
[20]Moschovakis, Y., Descriptive set theory, North Holland, Amsterdam, 1980.Google Scholar
[21]Shelah, S., Can you take Solovay's inaccessible away?, Israel Journal of Mathematics, vol. 48 (1984), pp. 1–47.CrossRefGoogle Scholar
[22]Shelah, S., On co-k-Souslin relations, Israel Journal of Mathematics, vol. 47 (1984), pp. 139–153.CrossRefGoogle Scholar
[23]Shelah, S. and Woodin, W., Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable, Israel Journal of Mathematics, vol. 70 (1990), pp. 381–393.CrossRefGoogle Scholar
[24]Solovay, R. M., On the cardinality of sets of reals, Symposium on Kurt Gödel (Buloff, et al., editors), 1969, pp. 58–73.Google Scholar
[25]Steel, J., The core model iterability problem, to appear in the Lecture Notes in Logic Series.Google Scholar
[26]Steel, J., HODL(ℝ) is a core model, Bulletin of Symbolic Logic, vol. 1 (1995), pp. 75–84.CrossRefGoogle Scholar