The finite and the infinite
The paper analyses the relation between various finiteness classes in ZF set theory without Choice. It starts with an an extensive background collecting citations of the most influential mathematicians and philosophers (Weyl, Hilbert, Russell, Pascal, Galileo, Leibniz, Bolzano, Dedekind, von Neumann) on the nature of infinity. Then, the author introduces a notion of finiteness class. This is a class of sets that contains all finite sets (i.e., sets that embed into the set \(\mathbb N\) of natural numbers), does not contain \(\mathbb N\), and is hereditary (in the sense that together with any set \(X\) it contains all sets that embed injectively into \(X\)). In Theorems 9 and 10, the author studies the relation between five finiteness classes: finite sets; A-finite sets (those that cannot be written as the disjoint union of two infinite sets); B-finite sets (those that contain no infinite linearly orderable subset); C-finite sets (those that cannot be mapped onto \(\mathbb N\)); and D-finite sets (those that contain no copy of \(\mathbb N\)). It is shown that the classes of finite sets and of D-finite sets are the smallest and the largest finiteness classes, respectively. In Theorem 12, sixteen conditions equivalent to the existence of a unique finiteness class are given. In Theorem 22 the author compares the five finiteness classes by their properties: coheredity; (finite) summability; (finite) productivity; power-stability. The paper ends with three open problems. The first of them asks whether each C-finite set is B-finite.
scientific article; zbMATH DE number 5886919