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.