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ITERATING SYMMETRIC EXTENSIONS

Published online by Cambridge University Press:  14 March 2019

ASAF KARAGILA*
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM EDMOND J. SAFRA CAMPUS, GIVAT RAM JERUSALEM91904, ISRAEL and SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIA NORWICH NR4 7TJ, UK E-mail: karagila@math.huji.ac.ilURL: http://karagila.org

Abstract

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Blass, A., Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society, vol. 255 (1979), pp. 3159.10.1090/S0002-9947-1979-0542870-6CrossRefGoogle Scholar
Feferman, S., Some applications of the notions of forcing and generic sets. Fundamenta Mathematicae, vol. 56 (1964), pp. 325345.10.4064/fm-56-3-325-345CrossRefGoogle Scholar
Grigorieff, S., Intermediate submodels and generic extensions in set theory. Annals of Mathematics(2), vol. 101 (1975), pp. 447490.10.2307/1970935CrossRefGoogle Scholar
Halpern, J. D. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic Set Theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. XIII, part I, American Mathematical Society, Providence, RI, 1971, pp. 83134.10.1090/pspum/013.1/0284328CrossRefGoogle 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, 1973.Google Scholar
Karagila, A., The Bristol model: An abyss called a Cohen real. Journal of Mathematical Logic, vol. 18 (2018), no. 2, 1850008.10.1142/S0219061318500083CrossRefGoogle Scholar
Kinna, W. and Wagner, K., Über eine Abschwächung des Auswahlpostulates. Fundamenta Mathematicae, vol. 42 (1955), pp. 7582.10.4064/fm-42-1-75-82CrossRefGoogle Scholar
Monro, G. P., Models of ZF with the same sets of sets of ordinals. Fundamenta Mathematicae , vol. 80 (1973), no. 2, pp. 105110.10.4064/fm-80-2-105-110CrossRefGoogle Scholar
Pincus, D., The dense linear ordering principle, this Journal, vol. 62 (1997), no. 2, pp. 438456.Google Scholar
Sageev, G., An independence result concerning the axiom of choice. Annals of Mathematical Logic, vol. 8 (1975), pp. 1184.10.1016/0003-4843(75)90002-9CrossRefGoogle Scholar
Sageev, G., A model of ZF+ there exists an inaccessible, in which the Dedekind cardinals constitute a natural nonstandard model of arithmetic. Annals of Mathematical Logic, vol. 21 (1981), no. 2–3, pp. 221281.10.1016/0003-4843(81)90017-6CrossRefGoogle Scholar
Shani, A., Borel reducibility and symmetric models, preprint, 2018, arXiv:1810.06722.Google Scholar