Category:Definitions/Zermelo Set Theory
This category contains definitions related to Zermelo Set Theory.
Related results can be found in Category:Zermelo Set Theory.
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
- 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:
- $\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$
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