# Definition:Zermelo-Fraenkel Set Theory with Axiom of Choice

## Definition

**Zermelo-Fraenkel Set Theory with the Axiom of Choice** 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 and the (controversial) Axiom of Choice.

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

- 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

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

### 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:

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 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.

## Also known as

**Zermelo-Fraenkel Set Theory with the Axiom of Choice** is popularly seen abbreviated as **ZFC**.

## Also see

## Source of Name

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

## Historical Note

**ZFC** is generally accepted by mathematicians as a "reasonably good foundation" of mathematics.

*We are far from claiming superiority of***ZFC**over alternative foundations of mathematics. For whatever reason, it won the competition. It does a decent job; so let us stick to it. It should be pointed out though that, to the best of our knowledge, none of the competitors to**ZFC**resolves the question of truth or falsity of CH, SH, MA, $\diamondsuit$, or of any other statement whose independence of**ZFC**has been established by the method of forcing.- -- 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*: Introduction

- -- 1996: Winfried Just and Martin Weese:

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Zermelo-Fraenkel set theory**