Definition 1

A set is said to be infinite if it is not finite.

In particular, a cardinal is infinite if it is not an integer.

The relation "there exists an infinite set" implies that the relation "$x$ is an integer" is collectivizing (Chapter II, § 1, no. 4); for if $\mathfrak{a}$ is an infinite cardinal and $n$ an arbitrary integer, we cannot have $\mathfrak{a} \leqslant n$ (§ 4, no. 2, Proposition 2). We have therefore $n < \mathfrak{a}$ for all integers $n$, which shows that the set of integers $< \mathfrak{a}$ (§ 3, no. 2, Remark following Theorem 1) contains all integers. Conversely, if the relation "$x$ is an integer" is collectivizing, then the set E of integers is an infinite set. For each integer $n$, the interval $[0, n]$ is a subset of $n + 1$ elements of E (§ 5, no. 3, Proposition 5). Therefore Card (E) $\geqslant n + 1 > n$. But to say that Card(E) $\neq n$ for every integer $n$ means that E is infinite.

¶ We now introduce the following axiom :

A5 ("Axiom of infinity".) There exists an infinite set.

It is not known whether or not this axiom can be deduced from the axioms and axiom schemes introduced previously; although the question has not been definitively settled, it is to be presumed that this axiom is independent of the others.

The preceding remarks then prove the following theorem :