The Axiom of Infinity

The axiom of infinity is the most profound of the ZFC axioms, as it asserts the existence of an actually infinite set. Without this axiom, we could only prove the existence of finite sets, making much of modern mathematics impossible.

DefinitionAxiom of Infinity

There exists a set $I$ such that $\emptyset \in I$ and whenever $x \in I$ , we also have $x \cup \{x\} \in I$ . Formally:

\exists I(\emptyset \in I \land \forall x(x \in I \rightarrow x \cup \{x\} \in I))

Such a set $I$ is called an inductive set.

The axiom of infinity provides the foundation for arithmetic and analysis. It guarantees that we can perform the successor operation $x \mapsto x \cup \{x\}$ infinitely many times, creating an endless sequence of distinct sets.

ExampleConstruction of Natural Numbers

Using the von Neumann construction, we define natural numbers as:

\begin{align*} 0 &= \emptyset \\ 1 &= 0 \cup \{0\} = \{\emptyset\} = \{0\} \\ 2 &= 1 \cup \{1\} = \{0, 1\} \\ 3 &= 2 \cup \{2\} = \{0, 1, 2\} \\ &\vdots \end{align*}

In general, each natural number $n$ is the set of all smaller natural numbers: $n = \{0, 1, 2, \ldots, n-1\}$ .

The successor function $S(x) = x \cup \{x\}$ plays a central role. For natural numbers, $S(n) = n \cup \{n\}$ corresponds to adding one. This definition ensures that each number contains exactly as many elements as its value, and that the membership relation $\in$ corresponds to the less-than relation $<$ on natural numbers.

DefinitionThe Set of Natural Numbers

The set of natural numbers $\omega$ (or $\mathbb{N}$ ) is the smallest inductive set, defined as the intersection of all inductive sets:

\omega = \bigcap \{I : I \text{ is inductive}\}

This set exists by the axiom of infinity and the axiom schema of comprehension.

The smallness condition is crucial: $\omega$ contains exactly those sets that must be in every inductive set. This means $\omega$ contains $0, 1, 2, 3, \ldots$ and nothing else. In particular, $\omega$ does not contain itself, and it does not contain any infinite sets other than those constructed from natural numbers.

Remark

The axiom of infinity is independent of the other axioms of ZFC. Without it, we can construct finite set theory (FST), where only finite sets exist. FST is consistent if ZFC is, but it cannot prove the existence of $\mathbb{N}$ , $\mathbb{R}$ , or most mathematical structures studied in analysis and topology.

The existence of $\omega$ has profound implications. It allows us to define:

Arithmetic: Addition and multiplication can be defined via recursion on $\omega$
Recursion: The principle of definition by recursion relies on $\omega$
Induction: Mathematical induction becomes a theorem about $\omega$
Sequences: Infinite sequences are functions from $\omega$ to other sets

Moreover, $\omega$ is the first example of a transfinite set—a set that is equinumerous with a proper subset of itself. Specifically, $\omega \sim \omega \setminus \{0\}$ via the function $n \mapsto n+1$ .

ExampleProperties of omega

The set $\omega$ has several remarkable properties:

Well-ordered: Every non-empty subset of $\omega$ has a least element
Transitive: If $n \in \omega$ and $m \in n$ , then $m \in \omega$
Not self-membered: $\omega \notin \omega$
Limit ordinal: $\omega$ is not the successor of any natural number

These properties make $\omega$ the prototype for all infinite ordinal numbers.

The axiom of infinity opens the door to infinite mathematics. Once we have $\omega$ , we can construct $\mathcal{P}(\omega)$ (the real numbers), $\mathcal{P}(\mathcal{P}(\omega))$ , and so on, climbing through an infinite hierarchy of ever-larger infinities. This hierarchy, formalized through the theory of ordinal and cardinal numbers, reveals the rich structure of infinity that ZFC set theory makes rigorous.