Zermelo set theory

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The Zermelo set theory is the first published axiomatic set theory ; it comes from Ernst Zermelo and is dated July 30, 1907. It was published on February 13, 1908 in volume 65 (2nd issue) of the Mathematische Annalen under the title Investigations on the Fundamentals of Set Theory and is the basis of Zermelo- Fraenkel set theory , which today serves as the basis of mathematics.

In order to give set theory a solid formal basis, Bertrand Russell published his type theory in 1903 , which, however, was difficult to access due to its syntactically complex form. Zermelo therefore chose the more elegant way of the axiomatic structure of set theory. His seven set axioms, which above all ensure the existence of sets, proved to be viable and, in the expanded form of Zermelo-Fraenkel set theory, allow the complete derivation of Cantor's set theory . Zermelo still formulated his axioms verbally; today, however, they are mostly specified in the form of predicate logic.

Zermelos Axioms 1907

Zermelo formulated his seven axioms for a range of things that contains the quantities as a sub-range. Namely, he defined sets as things containing elements or the zero set (empty set). However, the system of axioms also allows other elementless things as elements, which he later called primordial elements . You can equate elements and things with him, but not quantities and elements. In the original naming and numbering and in the original verbal wording, which in the following only leaves out commenting insertions with synonymous formulations, his axioms are:

I. Axiom of Determination:

If every element of a set M is at the same time an element of N and vice versa, then M = N.

II. Axiom of Elementary Sets:

There is a set, the zero set 0, which does not contain any elements.
If a is any thing of the domain, then there exists a set {a} which contains a and only a as an element.
If a, b are any two things in the domain, then there always exists a set {a, b} which contains both a and b, but no different thing x as an element.

III. Axiom of Separation:

If the class statement E (x) is definite for all elements of a set M, then M always has a subset which contains all those elements x of M for which E (x) is true, and only those elements as elements.

IV. Axiom of the power set:

Every set T corresponds to a second set UT, which contains all subsets of T and only those as elements.

V. Axiom of Association:

Every set T corresponds to a set ST, which contains all elements of the elements of T and only those as elements.

VI. Axiom of choice:

If T is a set whose elements are different from 0 and are foreign to each other, then its union ST contains at least one subset which has one and only one element in common with each element of T.

VII. Axiom of the Infinite:

The area contains at least one set Z, which contains the zero set as an element and is such that each of its elements a corresponds to a further element of the form {a}.

The axiom of infinity requires an inductive set (closed with respect to the enumeration a + 1 = { a }). Subsequently, Zermelo gave the first precise explicit definition of the natural numbers as the smallest set Z that satisfies the axiom of infinity. With this definition all Peano axioms are provable and the proof principle of complete induction .

The axiom system is slightly redundant, because the elementary set 0 can be obtained by separating out of the infinite set Z with the class statement x ≠ x and the elementary set {a} can be defined by the pair set {a, a}. So you only need the third elementary set {a, b}.

Original ZF system 1930

In an essay from 1930, Zermelo expanded his set axioms from 1907. He added to the replacement axiom that Abraham Fraenkel introduced in 1921 for the complete derivation of Cantor's set theory, and eliminated the two dispensable elementary sets that Fraenkel had derived. He gave this modified axiom system the name "Zermelo-Fraenkel system" or "ZF system". He gave Fraenkel's axiom the following wording:

Axiom of Replacement:

If the elements x of a set are uniquely replaced by any elements x 'of the range, then this contains a set m', which has all these x 'elements.

The axiom of substitution means that images of sets are also sets. Zermelo pointed out that the extended system of axioms is redundant: the axiom of discarding is provable with the axiom of substitution, and the elementary set can be derived from the power set and the null set using the axiom of substitution (because {a, b} is the image of the double Power set of null set). So he already knew an optimized ZF axiom system that gets by with the Zermelo axioms I, VI, V, VII, VIII and the substitution.

Zermelo's essay was actually about his general set theory, his "supplemented ZF system" or "ZF 'system". Here he left out the infinity axiom, exchanged the axiom of choice for the well-order theorem, and added the axiom of foundation that excludes circular sets, including all sets that contain themselves as elements. He formulated it for any things in the field, including primitive elements.

Axiom of foundation (second formulation Zermelos):

Each sub-area T contains at least one element t 0 that has no element t in T.

Modified ZF systems

Later formalized ZF systems differ from the original in several ways:

  • They eliminate Zermelo's framework with things and primordial elements and are pure set theory in which all objects are sets, which is achieved through a stronger axiom of determinateness (extensionality).
  • You do not count the axiom of choice as ZF and call the complete system with choice ZFC (C = Choice).
  • Since Skolem's suggestion of 1922, they have been using a formal language based on predicate logic that differs greatly from Zermelo's wording. He himself used a class logic based on Richard Dedekind , Giuseppe Peano and Ernst Schröder .
  • His counting in the axiom of infinity with is mostly replaced by his later counting from the set theory of 1930.


Individual evidence

  1. Skolem: Some remarks on the axiomatic justification of set theory, 1922, in: selected works, Oslo 1970, pp. 137–152.
  2. Zermelo: Investigations on the basics of set theory, p. 261ff.