# Definition:Zermelo Set Theory

## Contents

## Definition

**Zermelo set theory** is a system of axiomatic set theory.

Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the following axioms:

### 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

There exists a set that has no elements:

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

### 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

### Set Theory

There exists a set containing:

That is:

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

### Class Theory

Let $\omega$ be the class of natural numbers as constructed by the Von Neumann construction:

\(\displaystyle 0\) | \(:=\) | \(\displaystyle \O\) | |||||||||||

\(\displaystyle 1\) | \(:=\) | \(\displaystyle 0 \cup \set 0\) | |||||||||||

\(\displaystyle 2\) | \(:=\) | \(\displaystyle 1 \cup \set 1\) | |||||||||||

\(\displaystyle 3\) | \(:=\) | \(\displaystyle 2 \cup \set 2\) | |||||||||||

\(\displaystyle \) | \(\vdots\) | \(\displaystyle \) | |||||||||||

\(\displaystyle n + 1\) | \(:=\) | \(\displaystyle n \cup \set n\) | |||||||||||

\(\displaystyle \) | \(\vdots\) | \(\displaystyle \) |

Then $\omega$ is a set.

## Also see

- Results about
**Zermelo set theory**can be found here.

## Source of Name

This entry was named for Ernst Friedrich Ferdinand Zermelo.

## Historical Note

The axiomatic system of **Zermelo Set Theory** was created by Ernst Friedrich Ferdinand Zermelo as way to circumvent the logical inconsistencies of Frege set theory.

The **axiom of specification** was derived from the **comprehension principle**, with a domain strictly limited to the elements of a given pre-existing set.

Further axioms were then developed in order to allow the creation of such pre-existing sets:

- the axiom of existence, allowing for the existence of $\O := \set {}$
- the axiom of pairing, allowing for $\set {a, b}$ given the existence of $a$ and $b$
- the axiom of unions, allowing for $\bigcup a$ given the existence of a set $a$ of sets
- the axiom of powers, allowing for the power set $\powerset a$ to be generated for any set $a$
- the axiom of infinity, allowing for the creation of the set of natural numbers $\N$.

## 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