Journal of Symbolic Logic

On the prewellorderings associated with the directed systems of mice

Grigor Sargsyan

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Abstract

Working under $AD$, we investigate the length of prewellorderings given by the iterates of $\mathcal{M}_{2k+1}$, 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]).

Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 735-763.

Dates
First available in Project Euclid: 6 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1389032273

Digital Object Identifier
doi:10.2178/jsl.7803030

Mathematical Reviews number (MathSciNet)
MR3135496

Zentralblatt MATH identifier
1315.03087

Subjects
Primary: 03E15, 03E45, 03E60.

Keywords
Mouse Game Quantifier Prewellorderings Projective Ordinals Woodin cardinals

Citation

Sargsyan, Grigor. On the prewellorderings associated with the directed systems of mice. J. Symbolic Logic 78 (2013), no. 3, 735--763. doi:10.2178/jsl.7803030. https://projecteuclid.org/euclid.jsl/1389032273


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References

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