Zermelo set theory

(mathematics)

Zermelo set theory - 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. 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.