Zermelo-Fraenkel Set Theory

Chapter 6
Cardinals
6.1 Definition
The cardinal (cardinal number) of a set A is an object |A| such that the following equivalence
is satisfied:
|A| = |B| ⇔ A = 1 B.
(A = 1 B means that a bijection between A and B exists —See Definition 4.23.2 p. 35.)
A cardinal (number) is a cardinal (number) of a set.
The Frege-Russell definition of cardinals is |A| = def {B | B = 1 A}. This satisfies the
equivalence. However, |A| is a set only if A = ∅. Assuming AC, the following is a definition
of |A| as a set such that the above equivalence is provable:
Definition 6.1 |A| = def
⋂ {α ∈ OR | α =1 A}.
□
This defines |A| to be a canonical element of the Frege-Russell cardinal of A.
If the Regularity Axiom is available, the following definition can be used also (Scott;
see Lemma 4.22 p.32):
Definition 6.2 |A| = def Bottom({B | B = 1 A})
(= {B | B = 1 A ∧ ∀C (C = 1 A ⇒ ρ(B) ρ(C))}). □
This makes |A| a canonical selection of the Frege-Russell cardinal of A. If neither AC nor
Foundation are available, a definition of cardinality satisfying the required equivalence is
not possible (Levy). In that situation one solution remains: consider the operation | | as
a primitive notion and the above equivalence as an axiom.
For the following, it is irrelevant how cardinals have been introduced, as long as this
equivalence is satisfied.
6.2 Elementary Properties and Arithmetic
Definition 6.3 ℵ α = def |ω α |; an aleph is a cardinal of the form ℵ α .
□
AC is equivalent with the statement that every cardinal is an aleph.
Without loss of generality you may assume that 6.1 is satisfied whenever A has a
well-ordering. In that case:
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