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Models in Which AC Fails

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Combinatorial Set Theory

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Abstract

In Chap. 8 we have constructed models of Set Theory in which the Axiom of Choice failed. However, these models were models of Set Theory with atoms, denoted ZFA, where atoms are objects which do not have any elements but are distinct from the empty set. In this chapter we shall demonstrate how one can construct models of Zermelo–Fraenkel Set Theory (i.e., models of ZF) in which AC fails. Moreover, we shall also see how we can embed arbitrarily large fragments of permutation models (i.e., models of ZFA) into models of ZF.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Lorenz J. Halbeisen

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  1. Lorenz J. Halbeisen
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Halbeisen, L.J. (2017). Models in Which AC Fails. In: Combinatorial Set Theory. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-60231-8_17

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