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 (X _i )_i ∈ I of sets, indexed by a Dedekind-finite set I (i.e., there exists a family (Y _i )_i ∈ I of pairwise disjoint subsets Y _i of X _i 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

  $$r < s \Longrightarrow {\mathfrak{A}}_r \subsetneqq {\mathfrak{A}}_s.$$