The Axiom of Choice, the Generalized Continuum Hypothesis and Cardinal Arithmetic

Abstract

In Section 10 we defined the cardinal number of a set,

$$ \overline{\overline a} $$

, to be the smallest ordinal that is equivalent to a. If no such ordinal exists then

$$ \overline{\overline a} = 0 $$

. This definition has the advantage of connecting the theory of cardinal numbers to the properties of ordinals. A more traditional view is that

$$ \overline{\overline a} $$

is the equivalence class of sets equipollant to a.