Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T14:29:06.127Z Has data issue: false hasContentIssue false
  • English
  • Français

MINIMUM MODELS OF SECOND-ORDER SET THEORIES

Published online by Cambridge University Press:  08 April 2019

KAMERYN J. WILLIAMS
KAMERYN J. WILLIAMS*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAWAI‘I AT MĀNOA 2565 MCCARTHY MALL, KELLER 401A HONOLULU, HI 96822, USAE-mail: [email protected]: http://kamerynjw.net
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR.

Keywords

Primary 03E70Secondary 03C62Kelley–MorseGödel–Bernayselementary transfinite recursionminimum modelsecond-order set theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 589 - 620
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

Antos, C., Barton, N., and Friedman, S.-D., Universism and extensions of V, submitted.Google Scholar
Antos, C. and Friedman, S.-D., Hyperclass forcing in Morse-Kelley class theory, this Journal, vol. 82 (2017), no. 2, pp. 549–575.Google Scholar
Barwise, J., Admissible Sets and Structures: An Approach to Definability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975.Google Scholar
Cohen, P., A minimal model for set theory. Bulletin of the American Mathematical Society, vol. 69 (1963), no. 4, pp. 537–540.Google Scholar
Easton, W. B., Powers of regular cardinals, Ph.D. thesis, Princeton University, 1964.Google Scholar
Easton, W. B., Powers of regular cardinals. Annals of Mathematical Logic, vol. 1 (1970), no. 2, pp. 139–178.Google Scholar
Enayat, A., Models of set theory with definable ordinals. Archive for Mathematical Logic, vol. 44 (2005), pp. 363–385.Google Scholar
Felgner, U., Comparisons of the axioms of local and universal choice. Fundamenta Mathematicae, vol. 71 (1971), pp. 43–62.Google Scholar
Felgner, U., Choice functions on sets and classes, Sets and Classes on the Work by Paul Bernays (Müller, G. H., editor), Studies in Logic and the Foundations of Mathematics, vol. 84, Elsevier, Amsterdam, 1976, pp. 217–255.Google Scholar
Fujimoto, K., Classes and truths in set theory. Annals of Pure & Applied Logic, vol. 163 (2012), no. 11, pp. 1484–1523.Google Scholar
Gaifman, H., Global and local choice functions. Israel Journal of Mathematics, vol. 22 (1975), no. 3–4, pp. 257–265.Google Scholar
Gitman, V. and Hamkins, J. D., Open determinacy for class games, Foundations of Mathematics, Logic at Harvard, Essays in Honor of Hugh Woodin’s 60th Birthday (Caicedo, A. E., Cummings, J., Koellner, P., and Larson, P., editors), Contemporary Mathematics, American Mathematical Society, Providence, RI, 2017, pp. 121–143.Google Scholar
Gitman, V., Hamkins, J. D., Holy, P., Schlicht, P., and Williams, K. The exact strength of the class forcing theorem, manuscript, 2017.Google Scholar
Hamkins, J. D., Does ZFC prove the universe is linearly orderable? MathOverflow answer, 2012. Available at http://mathoverflow.net/q/110823(version: 2012-11-03).Google Scholar
Hamkins, J. D., Every countable model of set theory embeds into its own constructible universe. Journal of Mathematical Logic, vol. 13 (2013), no. 2, pp. 1350006, 27.Google Scholar
Hamkins, J. D., Linetsky, D., and Reitz, J., Pointwise definable models of set theory, this Journal, vol. 78 (2013), no. 1, pp. 139–156.Google Scholar
Harrison, J., Recursive pseudo-well orderings. Transactions of the American Mathematical Society, vol. 131 (1968), no. 2, pp. 526–543.Google Scholar
Jech, T., Set Theory, third ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
Kanamori, A., The Higher Infinite, corrected second edition, Springer-Verlag, Berlin, 2004.Google Scholar
J Keisler, H., Models with tree structures, Tarski Symposium: Proceedings of an International Symposium to Honor Alfred Tarski, Proceedings of Symposia in Pure Mathematics, vol. 25 (Henkin, L., editor), American Mathematical Society, Providence, RI, 1974, pp. 331–348.Google Scholar
Kossak, R. and Schmerl, J. H., The Structure of Models of Peano Arithmetic, Oxford Logic Guides, vol. 50, Oxford Science Publications, The Clarendon Press Oxford University Press, Oxford, 2006.Google Scholar
Marek, W. and Mostowski, A., On extendability of models of ZF set theory to the models of Kelley-Morse theory of classes, ISILC Logic Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel 1974, (Müller, G. H., Oberschelp, A., and Potthoff, K., editors), Springer Berlin Heidelberg, Berlin, Heidelberg, 1975, pp. 460–542.Google Scholar
Mitchell, W. J. and Steel, J. R., Fine Structure and Iteration Trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin, 1994.Google Scholar
Paris, J., Minimal models of ZF, Proceedings of the Bertrand Russell Memorial Conference (Bell, J. L., editor), Leeds University Press, Leeds, 1973, pp. 327–331.Google Scholar
Quinsey, J. E., Some problems in logic, Ph.D. thesis, University of Oxford, 1980.Google Scholar
Ratajczyk, Z., On sentences provable in impredicative extensions of theories. Dissertationes Mathematicae, vol. 178 (1979), p. 40.Google Scholar
Sato, K., Relative predicativity and dependent recursion in second-order set theory and higher order theories, this Journal, vol. 79 (2014), no. 3, pp. 712–732.Google Scholar
Scott, D. S., Measurable cardinals and constructible sets. Bulletin of the Polish Academy of Sciences, Mathematics, vol. 9 (1961), pp. 521–524.Google Scholar
Shelah, S., Models with second order properties ii. trees with no undefined branches. Annals of Mathematical Logic, vol. 14 (1978), no. 1, pp. 73–87.Google Scholar
Shepherdson, J. C., Inner models for set theory–part III, this Journal, vol. 18 (1953), no. 2, pp. 145–167.Google Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Logic, Association for Symbolic Logic, New York, 2009.Google Scholar
Williams, K. J., The structure of models of second-order set theory, Ph.D. thesis, The Graduate Center, CUNY, 2018.Google Scholar
Zarach, A. M., Replacement Collection, Gödel ’96: Logical Foundations of Mathematics, Computer Science and Physics—Kurt Gödel’s Legacy, Brno, Czech Republic, August 1996, Proceedings, Lecture Notes in Logic, vol. 6, Springer-Verlag, Berlin, 1996, pp. 307–322.Google Scholar
Zeman, M., Inner Models and Large Cardinals, De Gruyter, Berlin, 2001.Google Scholar