Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T23:37:56.629Z Has data issue: false hasContentIssue false
  • English
  • Français

PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

Published online by Cambridge University Press:  09 January 2018

PHILIPP SCHLICHT
PHILIPP SCHLICHT*
Affiliation:
MATHEMATISCHES INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN GERMANYE-mail: schlicht@math.uni-bonn.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire space λλ, where λ is an uncountable cardinal with λ<λ= λ. In the first main theorem, we show that the perfect set property for all subsets of λλ that are definable from elements of λOrd is consistent relative to the existence of an inaccessible cardinal above λ. In the second main theorem, we introduce a Banach–Mazur type game of length λ and show that the determinacy of this game, for all subsets of λλ that are definable from elements of λOrd as winning conditions, is consistent relative to the existence of an inaccessible cardinal above λ. We further obtain some related results about definable functions on λλ and consequences of resurrection axioms for definable subsets of λλ.

Keywords

03E3503E6003E1503E15generalized descriptive set theoryperfect set propertyBanach–Mazur gamelong games
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1317 - 1355
Copyright
Copyright © The Association for Symbolic Logic 2017 

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

Abraham, U., Aronszajn trees on ${\aleph _2}$ and ${\aleph _3}$ .. Annals of Pure and Applied Logic, vol. 24 (1983), no. 3, pp. 213–230.CrossRefGoogle Scholar
Abraham, U. and Shelah, S., Forcing closed unbounded sets, this Journal, vol. 48 (1983), no. 3, pp. 643–657.Google Scholar
Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 775–883.Google Scholar
Friedman, S.-D., Hyttinen, T., and Kulikov, V., Generalized descriptive set theory and classification theory. Memoirs of the American Mathematical Society, vol. 230 (2014), no. 1081.Google Scholar
Friedman, S.-D., Khomskii, Y., and Kulikov, V., Regularity properties on the generalized reals. Annals of Pure and Applied Logic, vol. 167 (2016), no. 4, pp. 408–430.Google Scholar
Fuchs, G., Closed maximality principles: Implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276–308.Google Scholar
Hamkins, J. D. and Johnstone, T. A., Resurrection axioms and uplifting cardinals. Archive for Mathematical Logic, vol. 53 (2014), no. 3–4, pp. 463–485.Google Scholar
Halko, A. and Shelah, S., On strong measure zero subsets of κ 2. Fundamenta Mathematicae, vol. 170 (2001), no. 3, pp. 219–229.Google Scholar
Jech, T., Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
Kanamori, A., Perfect-set forcing for uncountable cardinals. Annals of Mathematical Logic, vol. 19 (1980), no. 1–2, pp. 97–114.Google Scholar
Kanamori, A., The Higher Infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2009.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.Google Scholar
Kovachev, B., Some Axioms of Weak Determinacy, Dissertation, Universität München, 2009.Google Scholar
Laguzzi, G., Generalized Silver and Miller measurability. Mathematical Logic Quarterly, vol. 61 (2015), no. 1–2, pp. 91–102.Google Scholar
Lücke, P., Ros, L. M., and Schlicht, P., The Hurewicz dichotomy for generalized Baire spaces, to appear, 2016.Google Scholar
Lücke, P. and Schlicht, P., Continuous images of closed sets in generalized Baire spaces. Israel Journal of Mathematics, vol. 209 (2015), no. 1, pp. 421–461.Google Scholar
Martin, D. A. and Steel, J. R., A proof of projective determinacy. Journal of the American Mathematical Society, vol. 2 (1989), no. 1, pp. 71–125.Google Scholar
Mekler, A. and Väänänen, J., Trees and ${\rm{\Pi }}_1^1$ -subsets of ω 1 ω 1, this Journal, vol. 58 (1993), no. 3, pp. 1052–1070.Google Scholar
Schlicht, P., Thin equivalence relations and inner models. Annals of Pure and Applied Logic, vol. 165 (2014), no. 10, pp. 1577–1625.Google Scholar
Shelah, S., Can you take Solovay’s inaccessible away? Israel Journal of Mathematics, vol. 48 (1984), no. 1, pp. 1–47.Google Scholar
Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable. The Annals of Mathematics (2), vol. 92 (1970), pp. 1–56.Google Scholar
Steel, J. R., An outline of inner model theory, Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 1595–1684.Google Scholar
Väänänen, J., A Cantor–Bendixson theorem for the space ω 1 ω 1 . Fundamenta Mathematicae, vol. 137 (1991), no. 3, pp. 187–199.Google Scholar