Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T04:19:14.241Z Has data issue: false hasContentIssue false

REALIZING REALIZABILITY RESULTS WITH CLASSICAL CONSTRUCTIONS

Published online by Cambridge University Press:  19 December 2019

ASAF KARAGILA*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIA NORWICH, NR4 7TJ, UK E-mail: karagila@math.huji.ac.ilURL: http://karagila.org

Abstract

J. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2020 

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

Banaschewski, B. and Moore, G. H., The dual Cantor-Bernstein theorem and the partition principle. Notre Dame Journal of Formal Logic, vol. 31 (1990), no. 3, pp. 375381.CrossRefGoogle Scholar
Blass, A. and Scedrov, A., Freyd’s models for the independence of the axiom of choice. Memoirs of the American Mathematical Society, vol. 79 (1989), no. 404.CrossRefGoogle Scholar
Fontanella, L. and Geoffroy, G., Preserving cardinals and weak forms of Zorn’s lemma in realizability models, preprint, 2019.Google Scholar
Grigorieff, S., Intermediate submodels and generic extensions in set theory. Annals of Mathematics (2), vol. 101 (1975), pp. 447490.CrossRefGoogle Scholar
Jech, T., Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
Jech, T. J., The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, vol. 75, North-Holland, Amsterdam-London; Amercan Elsevier, New York, 1973.Google Scholar
Karagila, A., Iterating symmetric extensions. Journal of Symbolic Logic, vol. 84 (2019), no. 1, pp. 123159.CrossRefGoogle Scholar
Karagila, A., Preserving dependent choice. Bulletin of the Polish Academy of Sciences Mathematics, vol. 67 (2019), no. 1, pp. 1929.CrossRefGoogle Scholar
Krivine, J.-L., Realizability algebras II: New models of ZF + DC. Logical Methods in Computer Science, vol. 8 (2012), no. 1–10, p. 28.CrossRefGoogle Scholar
Krivine, J.-L., Realizability algebras III: some examples. Mathematical Structures in Computer Science, vol. 28 (2018), no. 1, pp. 4576.CrossRefGoogle Scholar
Usuba, T., Choiceless Löwenheim-Skolem property and uniform definability of grounds, 2019, arXiv:1904.00895, submitted.Google Scholar