Zermelo set theory
<mathematics> 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)
Nearby terms:
ZENO « zepto « Zermelo Fränkel set theory «
Zermelo set theory » ZERO » zero » Zero and Add
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