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Forms of Choice

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Combinatorial Set Theory

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Abstract

In 1904, Zermelo published his first proof that every set can be well-ordered. The proof is based on the so-called Axiom of Choice, denoted AC, which, in Zermelo’s words, states that the product of an infinite totality of sets, each containing at least one element, itself differs from zero (i.e., the empty set). The full theory ZF + AC, denoted ZFC, is called Set Theory.

The Axiom of Choice—which completes the axiom system of Set Theory and which is in our counting the ninth axiom of ZFC—states as follows:

9. The Axiom of Choice

$$\displaystyle{\forall \mathscr{F}\Big(\emptyset \notin \mathscr{F} \rightarrow \exists \ f\Big(\,f \in ^{\mathrm{\mathscr{F}}}\bigcup \mathscr{F} \wedge \forall x \in \mathscr{ F}\big(\,f(x) \in x\big)\Big)\Big).}$$

Informally, every family of non-empty sets has a choice function, or equivalently, every Cartesian product of non-empty sets is non-empty.

In this chapter, the Axiom of Choice—as well as some weaker forms of it—will be discussed in great detail.

I will say, however, that there are many species of counterpoint and that when the same notes and intervals of the principal are sung in the inversion, there will result a striking change in the harmony. Though there are many ways of writing such counterpoints, as I have said, I shall demonstrate only those that seem most elegant. This will avoid boring the reader, who can readily infer the other procedures for himself.

Gioseffo Zarlino

Le Istitutioni Harmoniche, 1558

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

    Lorenz J. Halbeisen

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Halbeisen, L.J. (2017). Forms of Choice. In: Combinatorial Set Theory. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-60231-8_6

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