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
- 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.
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): Zermelo-Fraenkel set theory
- 2002: Thomas Jech: Set Theory (3rd ed.) ... (next): Chapter $1$: Axioms of Zermelo-Fraenkel
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): 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