The Finite and the Infinite

Abstract

In ZF set theory finiteness classes are introduced and their stability under basic set theoretical constructions are being investigated. Typical results are: 1. The class of finite sets is the smallest finiteness class. 2. The class of Dedekind-finite sets is the largest finiteness class. 3. The class of almost finite sets is the largest summable finiteness class. 4. Equivalent are: a. There is only one finiteness class. b. The union of each family of 1-element sets, indexed by a Dedekind-finite set, is Dedekind-finite. c. The axiom of choice, for countable families of Dedekind-finite sets. d. The shrinking principle for families (Xi)i ∈ I of sets, indexed by a Dedekind-finite set I (i.e., there exists a family (Yi)i ∈ I of pairwise disjoint subsets Yi of Xi with \(\underset{i\in I}{\bigcup} Y_i = \underset{i\in I}{\bigcup} X_i\)). 5. In suitable ZF-models there exist families \(({\mathfrak{A}}_r)_{r\in{\mathbb{R}}}\) of finiteness classes such that $$r < s \Longrightarrow {\mathfrak{A}}_r \subsetneqq {\mathfrak{A}}_s.$$