Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T13:08:52.440Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Prewellorderings Associated with the Directed Systems of Mice

Published online by Cambridge University Press:  12 August 2016

Grigor Sargsyan
Grigor Sargsyan*
Affiliation:
Department of Mathematics, University of California, Los Angeles, California, 90095, USA, URL: http://math.ucla.edu/~grigor, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Working under AD, we investigate the length of prewellorderings given by the iterates of ℳ2k+1, which is the minimal proper class mouse with 2k + 1 many Woodin cardinals. In particular, we answer some questions from [4] (the discussion of the questions appears in the last section of [2]).

Keywords

03E1503E4503E60MouseGame QuantifierPrewellorderingsProjective OrdinalsWoodin cardinals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 735 - 763
Copyright
Copyright © Association for Symbolic Logic 2013

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] Harrington, Leo A. and Kechris, Alexander S., On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), no. 2, pp. 109–154.Google Scholar
[2] Hjorth, Greg, A boundedness lemma for iterations, available at http://www.math.ucla.edu/~greg/research.html. Google Scholar
[3] Hjorth, Greg, Variations of the Martin-Solovay tree, this Journal, vol. 61 (1996), no. 1, pp. 40–51.Google Scholar
[4] Hjorth, Greg, A boundedness lemma for iterations, this Journal, vol. 66 (2001), no. 3, pp. 1058–1072.Google Scholar
[5] Jackson, Steve, Structural consenquences of AD, Handbook of set theory (Foreman, Matt and Kanamori, Akihiro, editors), vol. 3, Springer-Verlag, 2010, pp. 1753–1876.Google Scholar
[6] Kanamori, Akihiro, The higher infinite, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
[7] Kechris, A. S. and Martin, D. A., On the theory of sets of reals, Bulletin of the American Mathematical Society, vol. 84 (1978), no. 1, pp. 149–151.Google Scholar
[8] Kechris, Alexander S., Martin, Donald A., and Solovay, Robert M., Introduction to Q-theory, Cabal seminar 79-81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 199–282.Google Scholar
[9] Martin, Donald A., The largest countable this, that, and the other, Cabal seminar 79-81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 97–106.Google Scholar
[10] Martin, Donald A. and Steel, John R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 1–73.Google Scholar
[11] Mitchell, William J. and Steel, John R., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer, Berlin, 1994.Google Scholar
[12] Moschovakis, Yiannis N., Descriptive set theory, second ed., Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, Providence, RI, 2009.Google Scholar
[13] Itayneeman, , Optimal proofs of determinacy, The Bulletin of Symbolic Logic, vol. 1 (1995), no. 3, pp. 327–339.Google Scholar
[14] Itayneeman, , Optimal proofs of determinacy. II, Journal of Mathematical Logic, vol. 2 (2002), no. 2, pp. 227–258.Google Scholar
[15] Sargsyan, Grigior, A tale ofhybridmice, Available at http://math.ucla.edu/~grigor. Google Scholar
[16] Schindler, Ralf and Steel, John R., The core model induction, Available at http://math.berkeley.edu/~steel. Google Scholar
[17] Schindler, Ralf and Steel, John R., The self-iterability of L[E], this Journal, vol. 74 (2009), no. 3, pp. 751–779.Google Scholar
[18] Solovay, Robert M., A coding of the subsets of ωω , Cabal Seminar 76-77, Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 133–150.CrossRefGoogle Scholar
[19] Steel, John R., Projectively well-ordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), no. 1, pp. 77–104.CrossRefGoogle Scholar
[20] Steel, John R., HODL(R) is a core model below Θ, The Bulletin of Symbolic Logic, vol. 1 (1995), no. 1, pp. 75–84.Google Scholar
[21] Steel, John R., PFA implies ADL(ℝ), this Journal, vol. 70 (2005), no. 4, pp. 1255–1296.Google Scholar
[22] Steel, John R., An outline of inner model theory, Handbook of set theory (Foreman, M. and Kanamori, A., editors), vol. 3, Springer, 2010, pp. 1595–1684.Google Scholar
[23] Steel, John R., Woodin's analysis of HODL(ℝ), available at www.math.berkeley.edu/~steel.\ Google Scholar
[24] Zeman, Martin, Inner models and large cardinals, de Gruyter Series in Logic and its Applications, vol. 5, Walter de Gruyter & Co., Berlin, 2002.Google Scholar