Definition:Zermelo-Fraenkel Axioms

Definition

The Zermelo-Fraenkel axioms are the most well-known basis for axiomatic set theory.

There is no standard numbering for them, and their exact formulation varies.

Certain of these axioms can in fact be derived from other axioms, so their status as "axioms" can be questioned.

The axioms 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:

every element of $A$ is also an element of $B$

and:

every element of $B$ is also an element of $A$.

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

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

The Axiom of Pairing

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

$\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$

The Axiom of Specification

For any well-formed formula $\map P y$, 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:

$(1): \quad$ a set with no elements

$(2): \quad$ the successor of each of its elements.

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 every mapping $f$ and subset $S$ of the domain of $f$, there exists a set containing the image $f \sqbrk S$.

More formally, let us express this as follows:

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

That is, we have:

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

Then we state as an axiom:

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

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.

The above axioms taken together as a system, but without the Axiom of Choice below, is called Zermelo-Fraenkel set theory.

This is often seen abbreviated ZF.

The Axiom of Choice

For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set.

$\ds \forall s: \paren {\O \notin s \implies \exists \paren {f: s \to \bigcup s}: \forall t \in s: \map f t \in t}$

That is, one can always create a choice function for selecting one element from each element of the set.

Whether or not the Axiom of Choice (AC) is accepted is more or less a philosophical position.

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The system of ZF set theory in combination with the Axiom of Choice is known as ZFC set theory: ZF plus 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

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Zermelo-Fraenkel axioms (ZF axioms)
• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html