## 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(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?]

Last updated: 1995-03-30

### Nearby terms:

zepto ♦ Zermelo Fränkel set theory ♦ **Zermelo set theory** ♦ ZERO ♦ zero

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