Zermelo set theory
Zermelo set theory
(mathematics)Extensionality: two sets are equal if and only if they havethe 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 onefree variable and X a set then
x: x is in X and P.
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 excludingsets of elements with arbitrary properties - the Comprehensionaxiom only allows a property to be used to select elements ofan existing set.
Zermelo Fr?nkel set theory adds the Replacement axiom.