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Cardinal Relations in ZF Only

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Combinatorial Set Theory

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Abstract

In this chapter we shall compare different cardinal numbers in Zermelo–Fraenkel Set Theory, which is Set Theory without the Axiom of Choice. For example, it will be shown that for any infinite set A, the cardinality of the set of finite subsets of A is always strictly smaller than the cardinality of the power set of A.

To some it may appear novel that I include the fourth among the consonances, because practicing musicians have until now relegated it to the dissonances. Hence I must emphasise that the fourth is actually not a dissonance but a consonance.

Gioseffo Zarlino

Le Istitutioni Harmoniche, 1558

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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). Cardinal Relations in ZF Only. In: Combinatorial Set Theory. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-60231-8_5

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