 \(\ast 80\) Selections. A selection function for a class \(\kappa\) is a function \(f\) making each element \(x\) of \(\kappa\) to a member of \(x\). These are denoted by \( \in_{\Delta} `\kappa\). The cardinal number of the product of two classes \(\alpha X \beta\) is the cardinal number of the class of all pairs of members selected from \(\alpha\) and \(\beta\), so the guarantee that such selections exist is called the Multiplicative Axiom in PM. This is now known as the Axiom of Choice, which had been identified as an assumption used in proofs in set theory by Ernst Zermelo in 1904. In PM it is defined as asserting that if a class \(\kappa\) is a set of mutually exclusive, non-empty classes, then there exists a class \(\mu\) which contains exactly one member of each element of \(\kappa\).