# Definition:Zermelo-Fraenkel Set Theory

## Definition

**Zermelo-Fraenkel Set Theory** is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based.

Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of set theory.

These are as follows:

### The Axiom of Extension

Let $A$ and $B$ be sets.

The **axiom of extension** states that $A$ and $B$ are equal if and only if they contain the same elements.

That is, if and only if:

and:

This can be formulated as follows:

- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

### The Axiom of the Empty Set

### Formulation 1

There exists a set that has no elements:

- $\exists x: \forall y: \paren {\neg \paren {y \in x} }$

### Formulation 2

There exists a set for which membership leads to a contradiction:

- $\exists x: \forall y \in x: y \ne y$

### The Axiom of Pairing

For any two sets, there exists a set to which only those two sets are elements:

- $\forall A: \forall B: \exists x: \forall y: \paren {y \in x \iff y = A \lor y = B}$

Thus it is possible to create a set that contains as elements any two sets that have already been created.

### The Axiom of Specification

For any function of propositional logic $\map P y$, we introduce the axiom:

- $\forall z: \exists x: \forall y: \paren {y \in x \iff \paren {y \in z \land \map P y} }$

where each of $x$, $y$ and $z$ range over arbitrary sets.

### The Axiom of Unions

For every set of sets $A$, there exists a set $x$ (the **union** set) that contains all and only those elements that belong to at least one of the sets in the $A$:

- $\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$

### The Axiom of Powers

For every set, there exists a set of sets whose elements are all the subsets of the given set.

- $\forall x: \exists y: \paren {\forall z: \paren {z \in y \iff \paren {w \in z \implies w \in x} } }$

### The Axiom of Infinity

There exists a set containing:

That is:

- $\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$

### The Axiom of Replacement

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:

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

### The Axiom of Foundation

For all non-empty sets, there is an element of the set that shares no element with the set.

That is:

- $\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$

The antecedent states that $S$ is not empty.

## Also known as

**Zermelo-Fraenkel set theory** is often seen abbreviated as **ZF**.

## Also see

- Definition:ZFC, which is
**Zermelo-Fraenkel set theory**with the addition of the axiom of choice

## Source of Name

This entry was named for Ernst Friedrich Ferdinand Zermelo and Abraham Halevi Fraenkel.

## Historical Note

Ernst Zermelo first proposed this supposedly rigorous system of axiomatic set theory in $1900$, in order to confront the paradoxes which the axiom of comprehension lead to.

It was modified by Abraham Fraenkel in $1922$.

The system of **Zermelo-Fraenkel set theory** has formed the basis of most of the formulations of axiomatic set theory which have been created since.

## Sources

- 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): Preface to the First Edition - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Zermelo-Fraenkel set theory** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Zermelo-Fraenkel set theory** - 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