Definition:Zermelo Set Theory

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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:

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

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:

$(1): \quad$ a set with no elements
$(2): \quad$ the successor of each of its elements.

That is:

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


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