| Zermelo set theory A with the following set of s: 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 with one and X a set then {x: x is in X and P(x)} is a set. Infinity: There exists an . Power-set: If X is a set, so is its . 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) |
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