No Access

Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

Institute of Mathematics, University of Vienna, Augasse 2-6, UZA 1 - Building 2, 1090 Vienna, Austria

E-mail Address: mueller.sandra@univie.ac.at

Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, FRG, Germany

E-mail Address: rds@wwu.de

Search for more papers by this author

, and

W. Hugh Woodin

Department of Mathematics and Department of Philosophy, Harvard University, 2 Arrow Street Room 417, Cambridge, MA 02138, USA

E-mail Address: woodin@math.harvard.edu

Search for more papers by this author

https://doi.org/10.1142/S0219061319500132Cited by:8 (Source: Crossref)

Abstract

We prove the following result which is due to the third author. Let $n \geq 1$ $n \geq 1$ . If $Π_{n}^{1}$ $Π_{n}^{1}$ determinacy and $Π_{n + 1}^{1}$ $Π_{n + 1}^{1}$ determinacy both hold true and there is no $Σ_{n + 2}^{1}$ $Σ_{n + 2}^{1}$ -definable $ω_{1}$ $ω_{1}$ -sequence of pairwise distinct reals, then $M_{n}^{#}$ $M_{n}^{#}$ exists and is $ω_{1}$ $ω_{1}$ -iterable. The proof yields that $Π_{n + 1}^{1}$ $Π_{n + 1}^{1}$ determinacy implies that $M_{n}^{#} (x)$ $M_{n}^{#} (x)$ exists and is $ω_{1}$ $ω_{1}$ -iterable for all reals $x$ $x$ . A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$ $n \geq 1$ , namely the statement that $Π_{n + 1}^{1}$ $Π_{n + 1}^{1}$ determinacy implies $⅁^{(n)} (< ω^{2} - Π_{1}^{1})$ determinacy.

Keywords:

AMSC: 03E45, 03E55, 03E60, 03E35

References

1. M. Davis , Infinite games of perfect information, in Advances in Game Theory (Princeton University Press, 1964), pp. 85–101. Google Scholar
2. G. Fuchs, I. Neeman and R. Schindler , A criterion for coarse iterability, Arch. Math. Logic 49 (2010). Web of Science, Google Scholar
3. S. D. Friedman, R. Schindler and D. Schrittesser , Coding over core models, in Infinity, Computability, and Metamathematics, Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch, eds. S. Geschke, B. Löwe and P. Schlicht (Springer, 2014), pp. 167–182. Google Scholar
4. D. Gale and F. M. Stewart , Infinite games with perfect information, in Contributions to the Theory of Games, eds. H. W. Kuhn and A. W. Tucker , Vol. 2 (Princeton University Press, 1953), pp. 245–266. Google Scholar
5. L. Harrington , Analytic determinacy and $0^{#}$ , J. Symb. Logic 43 (1978) 685–693. Web of Science, Google Scholar
6. R. B. Jensen and J. R. Steel , K without the measurable, J. Symb. Logic 78(3) (2013) 708–734. Web of Science, Google Scholar
7. R. B. Jensen, Robust extenders, in Handwritten Notes (2003). Google Scholar
8. R. B. Jensen , The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972) 229–308. Google Scholar
9. R. B. Jensen , A new fine structure for higher core models, Circulated Manusc. (1997). Google Scholar
10. T. Jech , Set Theory, 3rd edn. (Springer-Verlag, Berlin, Heidelberg, 2003). Google Scholar
11. A. S. Kechris and Y. N. Moschovakis , Notes on the theory of scales, in Games, Scales and Suslin Cardinals, The Cabal Seminar, Volume I, eds. A. S. Kechris, B. Löwe and J. R. Steel (Cambridge University Press, 2008). Google Scholar
12. A. Kechris and R. Solovay , On the relative consistency strength of determinacy hypotheses, Trans. Amer. Math. Soc. 290(1) (1985). Web of Science, Google Scholar
13. A. S. Kechris and W. H. Woodin , The equivalence of partition properties and determinacy, in Games, Scales and Suslin Cardinals, The Cabal Seminar, Volume I, eds. A. S. Kechris, B. Löwe and J. R. Steel (Cambridge University Press, 2008). Google Scholar
14. P. Koellner and W. H. Woodin , Large cardinals from determinacy, in Handbook of Set Theory, eds. M. Foreman and A. Kanamori (Springer, 2010). Google Scholar
15. A. Kanamori , The Higher Infinite, 2nd edn. (Springer-Verlag, Berlin, Heidelberg, 2003). Google Scholar
16. A. S. Kechris , The perfect set theorem and definable wellorderings of the continuum, J. Symb. Logic 43(4) (1978) 630–634. Web of Science, Google Scholar
17. H. J. Keisler , Some applications of the theory of models to set theory, Proc. Int. Cong. Logic, Methodology, and Philosophy of Science (Springer, 1962), pp. 80–85. Google Scholar
18. P. B. Larson , The Stationary Tower: Notes on a Course by W. Hugh Woodin, University Lecture Series, Vol. 32 (AMS, 2004). Google Scholar
19. W. J. Mitchell and J. R. Steel , Fine Structure and Iteration Trees, Lecture Notes in Logic (Springer-Verlag, Berlin, New York, 1994). Google Scholar
20. W. J. Mitchell and R. Schindler , A universal extender model without large cardinals in $V$ , J. Symb. Logic 69(2) (2004) 371–386. Web of Science, Google Scholar
21. W. J. Mitchell and E. Schimmerling , Covering without countable closure, Math. Res. Lett. 2(5) (1995) 595–609. Web of Science, Google Scholar
22. W. J. Mitchell, E. Schimmerling and J. R. Steel , The covering lemma up to a Woodin cardinal, Ann. Pure Appl. Logic 84 (1997) 219–255. Web of Science, Google Scholar
23. D. A. Martin , The largest countable this, that, and the other, in Games, Scales and Suslin Cardinals, The Cabal Seminar, Volume I, eds. A. S. Kechris, B. Löwe and J. R. Steel (Cambridge University Press, 2008). Google Scholar
24. D. A. Martin , Measurable cardinals and analytic games, Fund. Math. 66 (1970) 287–291. Google Scholar
25. D. A. Martin , $Δ_{2 n}^{1}$ determinacy implies $Σ_{2 n}^{1}$ determinacy, Mimeograph. Notes (1973). Google Scholar
26. D. A. Martin , Borel determinacy, Ann. Math. 102(2) (1975) 363–371. Web of Science, Google Scholar
27. D. A. Martin and J. R. Steel , A proof of projective determinacy, J. Amer. Math. Soc. 2 (1989) 71–125. Google Scholar
28. D. A. Martin and J. R. Steel , Iteration trees, J. Amer. Math. Soc. 7(1) (1994) 1–73. Google Scholar
29. R. Mansfield , The non-existence of $Σ_{2}^{1}$ well-orderings of the Cantor set, Fund. Math. 86 (1975) 279–282. Google Scholar
30. Y. N. Moschovakis , Descriptive Set Theory, Mathematical Surveys and Monographs, Vol. 155, 2nd edn. (AMS, 2009). Google Scholar
31. Y. N. Moschovakis , Uniformization in a playful universe, Bull. Amer. Math. Soc. 77 (1971) 731–736. Web of Science, Google Scholar
32. J. Mycielski and S. Swierczkowski , On the Lebesgue measurability and the axiom of determinateness, Fund. Math. 54 (1964) 67–71. Google Scholar
33. I. Neeman , Optimal proofs of determinacy II, J. Math. Logic 2(2) (2002) 227–258. Link, Google Scholar
34. I. Neeman , Determinacy in $L (ℝ)$ , in Handbook of Set Theory, eds. M. Foreman and A. Kanamori (Springer, 2010). Google Scholar
35. I. Neeman , Optimal proofs of determinacy, Bull. Symb. Logic 1(3) (1995) 327–339. Web of Science, Google Scholar
36. G. Sargsyan , On the prewellorderings associated with the directed systems of mice, J. Symb. Logic 78(3) (2013) 735–763. Web of Science, Google Scholar
37. D. S. Scott , Measurable cardinals and constructible sets, Bull. Acad. Polon. Sci. 9 (1961) 521–524. Google Scholar
38. R. Schindler , Coding into $K$ by reasonable forcing, Trans. Amer. Math. Soc. 353 (2000) 479–489. Web of Science, Google Scholar
39. E. Schimmerling , A core model toolbox and guide, in Handbook of Set Theory, eds. M. Foreman and A. Kanamori (Springer, 2010). Google Scholar
40. R. Schindler , Set Theory, Universitext (Springer-Verlag, 2014). Google Scholar
41. R. Schindler and J. R. Steel, The core model induction. Google Scholar
42. R. Schindler and J. R. Steel , The self-iterability of $L [E]$ , J. Symb. Logic 74(3) (2009) 751–779. Web of Science, Google Scholar
43. R. Schindler, J. R. Steel and M. Zeman , Deconstructing inner model theory, J. Symb. Logic 67 (2002) 721–736. Web of Science, Google Scholar
44. F. Schlutzenberg and N. Trang, Scales in hybrid mice over $ℝ$ , Submitted (2016). Google Scholar
45. R. Schindler and M. Zeman , Fine structure, in Handbook of Set Theory, eds. M. Foreman and A. Kanamori (Springer, 2010). Google Scholar
46. R. M. Solovay, The cardinality of $Σ_{2}^{1}$ sets of reals, in Symposium Papers Commemorating the Sixtieth Birthday of Kurt Gödel, eds. J. J. Bulloff, T. C. Holyoke and S. W. Hahn (Springer, Berlin, Heidelberg, 1969), pp. 58–73. Google Scholar
47. J. R. Steel , An outline of inner model theory, in Handbook of Set Theory, eds. M. Foreman and A. Kanamori (Springer, 2010). Google Scholar
48. J. R. Steel , Inner models with many Woodin cardinals, Ann. Pure Appl. Logic 65 (1993) 185–209. Web of Science, Google Scholar
49. J. R. Steel , Projectively well-ordered inner models, Ann. Pure Appl. Logic 74 (1995). Web of Science, Google Scholar
50. J. R. Steel , The Core Model Iterability Problem, Lecture Notes in Logic, Vol. 8 (Springer-Verlag, Berlin, New York, 1996). Google Scholar
51. J. R. Steel and W. H. Woodin , HOD as a core model, in Ordinal Definability and Recursion Theory, The Cabal Seminar, Volume III, eds. A. S. Kechris, B. Löwe and J. R. Steel (Cambridge University Press, 2016). Google Scholar
52. P. Wolfe , The strict determinateness of certain infinite games, Pacific J. Math. 5 (1955) 841–847. Google Scholar
53. W. H. Woodin , On the consistency strength of projective uniformization, Proc. Herbrand Symp. Logic Colloquium’81, ed. J. Stern (North-Holland, 1982), pp. 365–383. Google Scholar