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What (who) is Zermelo set theory - definition
SYSTEM OF MATHEMATICAL SET THEORY
Axiom of elementary sets; Zermelo's axiom; Mac Lane set theory; MacLane set theory; Maclane set theory
Zermelo–Fraenkel set theory
STANDARD FORM OF AXIOMATIC SET THEORY
Zermelo-Fraenkel axiom; ZFC; Zermelo-Fraenkel axioms; Zermelo-Frankel axioms; Zermelo-Frankel set theory; ZFC set theory; Zermelo-Fraenkel framework; ZFC Set Theory; Zermelo-Fränkel set theory; Zermelo-Frankel; Zfc; ZFC set; ZF axioms; Zermelo–Frankel set theory; Zermelo-Fraenkel set theory; ZF set theory; Zermelo-Fraenkel-Skolem set theory; Zermelo-Frankel axiom; Zermelo–Fraenkel axioms; Zermelo Fraenkel set theory; Zermelo-Fraenkel; Zermelo–Fraenkel axiomatization; Zermelo-Fraenkel axiomatization; Zermelo-Fränkel; ZFC Set theory; Zermelo–Fraenkel axiom; Zermelo–Fraenkel framework; Zermelo–Fraenkel; Zermelo-frankel; Zermelo–Fraenkel set theory with the axiom of choice; Axioms of ZF; Zermelo−Fraenkel set theory; Zermelo-Fraenkel set theory with the axiom of choice
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.
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 Frankel set theory adds the Replacement axiom.
[Other axioms? ]
(1995-03-30)
Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted.
Wikipedia
Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
