# Axiom:Axiom of Replacement

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## Axiom

For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\map f S$.

More formally, let us express this as follows:

Let $\map P {x, z}$ be a propositional function, which determines a function.

That is, we have:

- $\forall x: \exists ! y : \map P {x, y}$.

Then we state as an axiom:

- $\forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x,y}}$

## Also presented as

The two above statements may be combined into a single (somewhat unwieldy) expression:

- $\forall x: \exists ! y : \map P {x, y} \implies \forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x,y}}$

## Also known as

The **axiom of replacement** is also known as the **axiom of substitution**.

## Historical Note

The **axiom of replacement** was added to the axioms of **Zermelo set theory** by Abraham Halevi Fraenkel, and also independently by Thoralf Albert Skolem.

The resulting system of axiomatic set theory is now referred to as **Zermelo-Fraenkel Set Theory**.

## Sources

- 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF7}$ - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Replacement Schema - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html