Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T15:16:02.920Z Has data issue: false hasContentIssue false
  • English
  • Français

$G_{\delta \sigma }$ GAMES AND INDUCTION ON REALS

Part of: Game theory Set theory Computability and recursion theory General logic

Published online by Cambridge University Press:  13 September 2021

J. P. AGUILERA  and
P. D. WELCH
J. P. AGUILERA
Affiliation:
DEPARTMENT OF MATHEMATICS GHENT UNIVERSITY KRIJGSLAAN 281-S8 B9000GHENTBELGIUM and INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY 1040 VIENNA, AUSTRIA E-mail: aguilera@logic.at
P. D. WELCH
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BRISTOLCLIFTON, BRISTOL BS8 1UG, UKE-mail: p.welch@bristol.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

Keywords

determinacygames on realsinductive definitions

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03D60: Computability and recursion theory on ordinals, admissible sets, etc. 03D70: Inductive definability 03E60: Determinacy principles 91A44: Games involving topology or set theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1676 - 1690
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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

Aguilera, J. P., Long Borel games. Israel Journal of Mathematics, vol. 243 (2021), pp. 273–314.CrossRefGoogle Scholar
Aguilera, J. P., Determined admissible sets. Proceedings of the American Mathematical Society, vol. 148 (2020), pp. 2217–2231.10.1090/proc/14914CrossRefGoogle Scholar
Aguilera, J. P., ${F}_{\sigma }$ games and reflection in $L\left(\mathbb{R}\right)$ , this Journal, vol. 85 (2020), pp. 1102–1123.Google Scholar
Aguilera, J. P. and Müller, S., Projective games on the reals. Notre Dame Journal of Formal Logic, vol. 64 (2020), pp. 573–589.Google Scholar
Barwise, J., Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975.10.1007/978-3-662-11035-5CrossRefGoogle Scholar
Blass, A., Equivalence of two strong forms of determinacy. Proceedings of the American Mathematical Society, vol. 52 (1975), pp. 373–376.10.1090/S0002-9939-1975-0373903-XCrossRefGoogle Scholar
Davis, M., Infinite games of perfect information, Advances in Game Theory, Princeton University Press, Princeton, 1964, pp. 85–101.Google Scholar
Friedman, H. M., Higher set theory and mathematical practice. Annals of Mathematics and Logic, vol. 2 (1971), no. 3, pp. 325 – 357.10.1016/0003-4843(71)90018-0CrossRefGoogle Scholar
Hachtman, S., Determinacy and monotone inductive definitions. Israel Journal of Mathematics, vol. 230 (2019), pp. 71–96.CrossRefGoogle Scholar
Jech, T. J., Set Theory, Springer Monographs in Mathematics. Springer, Berlin, 2003.Google Scholar
Kechris, A. and Solovay, R., On the relative consistency strength of determinacy hypotheses. Transactions of the American Mathematical Society, vol. 290 (1985), no. 1.Google Scholar
Kechris, A. S., Measure and category in effective descriptive set theory. Annals of Mathematics and Logic, vol. 5 (1973), no. 4, pp. 337–384.10.1016/0003-4843(73)90012-0CrossRefGoogle Scholar
Kechris, A. S. and Woodin, W. H., Equivalence of partition properties and determinacy. Proceedings of the National Academy of Sciences of the United States of America, vol. 80 (1983), pp. 1783–1786.10.1073/pnas.80.6.1783CrossRefGoogle ScholarPubMed
Martin, D. A., Borel determinacy. Annals of Mathematics, vol. 102 (1975), no. 2, pp. 363–371.CrossRefGoogle Scholar
Martin, D. A. and Steel, J. R., The extent of scales in $L\left(\mathbb{R}\right)$ , Games, Scales, and Suslin Cardinals. The Cabal Seminar, vol. I (Kechris, A. S., Löwe, B., and Steel, J. R., editors), The Association of Symbolic Logic, Cambridge University Press, New York, 2008, pp. 110–120.Google Scholar
Mathias, A. R. D., Provident sets and rudimentary set forcing. Fundamenta Mathematicae, vol. 230 (2015), pp. 99–148.CrossRefGoogle Scholar
Moschovakis, Y. N., Descriptive Set Theory, second ed., Mathematical Surveys and Monographs, 155, AMS, Providence, 2009.10.1090/surv/155CrossRefGoogle Scholar
Steel, J. R., Scales in $L\left(\mathbb{R}\right)$ . Games, Scales and Suslin Cardinals, The Cabal Seminar, vol. I (Kechris, A. S., Löwe, B., and Steel, J. R., editors), Cambridge University Press, Cambridge, 2008.Google Scholar
Welch, P. D., Weak systems of determinacy and arithmetical quasi-inductive definitions, this Journal, vol. 76 (2011), pp. 418–436.Google Scholar
Welch, P. D., ${G}_{\delta \sigma}$ -games. Isaac Newton Institute Pre-print series No. NI12050-SAS, July 2012. Available at https://people.maths.bris.ac.uk/~mapdw/.Google Scholar