Zermelo set theory

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Zermelo set theory
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{{No footnotes|date=April 2018}}Zermelo set theory (sometimes denoted by Z-), as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. 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.

The axioms of Zermelo set theory

The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are called sets, and the remaining objects are urelements and do not contain any elements. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary predicate.
AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M equiv N. Briefly, every set is determined by its elements." AXIOM II. Axiom of elementary sets (Axiom der Elementarmengen) "There exists a set, the null set, ∅, that contains no element at all. If a is any object of the domain, there exists a set {a} containing a and only a as an element. If a and b are any two objects of the domain, there always exists a set {a, b} containing as elements a and b but no object x distinct from them both." See Axiom of pairs. AXIOM III. Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is definite for all elements of a set M, M possesses a subset M'  containing as elements precisely those elements x of M for which –(x) is true." AXIOM IV. Axiom of the power set (Axiom der Potenzmenge) "To every set T there corresponds a set T' , the power set of T, that contains as elements precisely all subsets of T ." AXIOM V. Axiom of the union (Axiom der Vereinigung) "To every set T there corresponds a set ∪T, the union of T, that contains as elements precisely all elements of the elements of T ." AXIOM VI. Axiom of choice (Axiom der Auswahl) "If T is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ∪T includes at least one subset S1 having one and only one element in common with each element of T ." AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element."

Connection with standard set theory

The most widely used and accepted set theory is known as ZFC, which consists of Zermelo–Fraenkel set theory with the addition of the axiom of choice. The links show where the axioms of Zermelo's theory correspond. There is no exact match for "elementary sets". (It was later shown that the singleton set could be derived from what is now called "Axiom of pairs". If a exists, a and a exist, thus {a,a} exists. By extensionality {a,a} = {a}.) The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.Zermelo set theory does not include the axioms of replacement and regularity. The axiom of replacement was first published in 1922 by Abraham Fraenkel and Thoralf Skolem, who had independently discovered that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of natural numbers and Z'n+1 is the power set of Z'n. They both realized that the axiom of replacement is needed to prove this. The following year, John von Neumann pointed out that this axiom is necessary to build his theory of ordinals. The axiom of regularity was stated by von Neumann in 1925.Ferreirós 2007, pp. 369, 371.In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first order formula with parameters", so the separation axiom is replaced by an axiom scheme. The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive. Zermelo set theory is usually taken to be a first-order theory with the separation axiom replaced by an axiom scheme with an axiom for each first-order formula. It can also be considered as a theory in second-order logic, where now the separation axiom is just a single axiom. The second-order interpretation of Zermelo set theory is probably closer to Zermelo's own conception of it, and is stronger than the first-order interpretation.In the usual cumulative hierarchy Vα of ZFC set theory (for ordinals α), any one of the setsVα for α a limit ordinal larger than the first infinite ordinal ω (such as Vω·2) forms a model of Zermelo set theory. So the consistency of Zermelo set theory is a theorem of ZFC set theory. Zermelo's axioms do not imply the existence of ℵω or larger infinite cardinals, as the model Vω·2 does not contain such cardinals. (Cardinals have to be defined differently in Zermelo set theory, as the usual definition of cardinals and ordinals does not work very well: with the usual definition it is not even possible to prove the existence of the ordinal ω2.)The axiom of infinity is usually now modified to assert the existence of the first infinitevon Neumann ordinal omega; the original Zermeloaxioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo'saxiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of V_{omega} as a set nor of any rank of the cumulative hierarchy of sets with infinite index.Zermelo allowed for the existence of urelements that are not sets and contain no elements; these are now usually omitted from set theories.

Mac Lane set theory

Mac Lane set theory, introduced by {{harvs|txt|last=Mac Lane|year=1986|authorlink=Saunders Mac Lane}}, is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded,Mac Lane set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.

The aim of Zermelo's paper

The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "Russell antinomy".He says he wants to show how the original theory of Georg Cantor and Richard Dedekind can be reduced to a few definitions and seven principles or axioms. He says he has not been able to prove that the axioms are consistent.A non-constructivist argument for their consistency goes as follows. Define Vα for α one of the ordinals 0, 1, 2, ...,ω, ω+1, ω+2,..., ω·2 as follows:
  • V0 is the empty set.
  • For α a successor of the form β+1, Vα is defined to be the collection of all subsets of Vβ.
  • For α a limit (e.g. ω, ω·2) then Vα is defined to be the union of Vβ for β

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