Axiom Schema of Replacement

The axiom schema of replacement, introduced by Abraham Fraenkel and Thoralf Skolem, is one of the most powerful axioms in ZFC. It generalizes the axiom schema of comprehension and is essential for developing the theory of ordinal and cardinal numbers.

DefinitionAxiom Schema of Replacement

For any formula $\varphi(x, y, p_1, \ldots, p_n)$ in the language of set theory, if for every $x$ in a set $A$ there exists a unique $y$ such that $\varphi(x, y, p_1, \ldots, p_n)$ holds, then there exists a set $B$ containing exactly those $y$ values:

\forall A[\forall x \in A \, \exists! y \, \varphi(x, y, \vec{p}) \rightarrow \exists B \forall y(y \in B \leftrightarrow \exists x \in A \, \varphi(x, y, \vec{p}))]

where $\vec{p} = p_1, \ldots, p_n$ are parameters.

The replacement axiom asserts that if we have a functional relationship between sets (expressed by the formula $\varphi$ ), we can apply this function to all elements of a set $A$ to obtain a new set $B$ . This is a natural extension of the idea that functions map sets to sets.

ExampleTransfinite Recursion

Consider defining sets by recursion. Suppose we want to define a sequence of sets:

\begin{align*} A_0 &= \emptyset \\ A_{n+1} &= \mathcal{P}(A_n) \text{ for all } n \in \omega \end{align*}

Without replacement, we cannot prove that the collection $\{A_0, A_1, A_2, \ldots\}$ forms a set. Replacement guarantees that since the function $n \mapsto A_n$ is well-defined on $\omega$ , the image set exists.

The distinction between replacement and comprehension is subtle but crucial. Comprehension allows us to separate elements from an existing set based on a property. Replacement allows us to transform each element of a set and collect the results, even if those results were not originally members of any fixed set.

Remark

Zermelo's original axiomatization (1908) did not include replacement, only comprehension. This system, now called Zermelo set theory (Z), is weaker than ZFC. Many constructions in ordinal arithmetic, such as the operation $\alpha \mapsto \omega_\alpha$ , require replacement and cannot be carried out in Z.

Replacement has several equivalent formulations. One particularly useful version states that the image of a set under a class function (a function possibly too large to be a set) is again a set:

DefinitionClass Function Image

If $F$ is a class function (a formula defining a functional relationship) and $A$ is a set, then the image

F[A] = \{F(x) : x \in A\}

is also a set. This follows from replacement by taking $\varphi(x, y)$ to be " $y = F(x)$ ".

Applications of replacement are numerous in advanced set theory:

Ordinal arithmetic: Showing that $\alpha + \beta$ , $\alpha \cdot \beta$ , and $\alpha^\beta$ are ordinals
Cardinal operations: Defining infinite sums and products of cardinals
Cumulative hierarchy: Proving that $V_{\omega+\omega}$ exists, where $V_\alpha$ denotes the $\alpha$ -th level of the von Neumann hierarchy
Reflection principles: Deriving important metamathematical results about ZFC

ExampleBeth Numbers

The beth numbers form a hierarchy of cardinal numbers defined by:

\begin{align*} \beth_0 &= \aleph_0 = |\omega| \\ \beth_{\alpha+1} &= 2^{\beth_\alpha} \\ \beth_\lambda &= \sup_{\alpha < \lambda} \beth_\alpha \text{ for limit } \lambda \end{align*}

To prove that $\{\beth_\alpha : \alpha \in \text{Ord}\}$ forms a proper class rather than a set, we use replacement: if it were a set, we could apply replacement to obtain larger cardinals, contradicting its supposed maximality.

The power of replacement becomes especially apparent when working with large cardinals and infinitary combinatorics. For instance, the Hartogs number construction, which assigns to each set $X$ the least ordinal that does not inject into $X$ , relies fundamentally on replacement.

Remark

From a philosophical perspective, replacement embodies the idea that the set-theoretic universe is "closed under definable operations." If we can define a function in the language of set theory, and if that function's domain is a set, then its range must also be a set. This principle ensures that ZFC has sufficient closure properties to capture our intuitions about mathematical collections.

Without replacement, many natural mathematical constructions would be impossible to formalize. The axiom ensures that ZFC is robust enough to handle the transfinite iterations and abstract constructions that pervade modern mathematics, from functional analysis to category theory.