The Free On-line Dictionary of Computing (30 December 2018):

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?]

   (1995-03-30)