# 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 $\map P {y, z}$ be a propositional function, which determines a function.

That is, we have:

- $\forall y: \exists x: \forall z: \paren {\map P {y, z} \iff x = z}$.

Then we state as an axiom:

- $\forall w: \exists x: \forall y: \paren {y \in w \implies \paren {\forall z: \paren {\map P {y, z} \implies z \in x} } }$

## Also presented as

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

- $\paren {\forall y: \exists x: \forall z: \paren {\map P {y, z} \implies x = z} } \implies \forall w: \exists x: \forall y: \paren {y \in w \implies \forall z: \paren {\map P {y, z} \implies z \in x} }$

## 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

- 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