Zermelo set theory
<mathematics> A {set theory} with the following set of
{axioms}:
Extensionality: two sets are equal if and only if they have
the same elements.
Union: If U is a set, so is the union of all its elements.
Pair-set: If a and b are sets, so is
{a, b}.
Foundation: Every set contains a set disjoint from itself.
Comprehension (or Restriction): If P is a {formula} with one
{free variable} and X a set then
{x: x is in X and P(x)}.
is a set.
Infinity: There exists an {infinite set}.
Power-set: If X is a set, so is its {power set}.
Zermelo set theory avoids {Russell's paradox} by excluding
sets of elements with arbitrary properties - the Comprehension
axiom only allows a property to be used to select elements of
an existing set.
{Zermelo Fränkel set theory} adds the Replacement axiom.
[Other axioms?]
(1995-03-30)