Axiomatic set theory

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Any axiomatization of set theory that avoids the well-known antinomies of naive set theory counts as axiomatic set theory . The most common axiomatization in modern mathematics is the Zermelo-Fraenkel set theory with axiom of choice (ZFC). The oldest axiomatization comes from Georg Cantor , the founder of set theory.

History and characteristics

The first axiomatizations of set theory were attempted before the discovery of the set antinomies, namely in 1888 by Richard Dedekind and in 1893 by Gottlob Frege , both of which based arithmetic on a set calculus. But since both calculi have proven to be inconsistent because of axioms that allow unlimited set formation, they are counted as naive set theory. By axiomatic set theory one understands only those axiomatizations that try to avoid these contradictions of naive set theory by more restrictive set formation.

To avoid contradictions, Bertrand Russell proposed a step-by-step structure for set theory and developed his type theory from 1903 to 1908 , which also served as the basis for the 1910 Principia Mathematica . In it a multitude is always of a higher type than its elements. Among other things, the statement that a set contains itself as an element with which Russell's antinomy is formed cannot be formulated in this theory. The type theory tries to solve the problems through a restricted syntax of the admissible class statements. With Russell himself it does not yet have an axiomatic form, but was only later developed into a relatively complex axiomatic theory. Their consistency has been proven by Paul Lorenzen . The consistency of the Principia Mathematica, which is based on type theory, cannot be proven due to Gödel's incompleteness theorem . The type theory of Principia Mathematica was decisive in logic for a long time, but could not establish itself in mathematics practice, on the one hand because of its complexity and on the other hand because of its inadequacy. It is not enough to justify Cantor's set theory and mathematics, since their linguistic means are too weak.

Rather , the form of axiomatic set theory initiated by Ernst Zermelo gradually gained acceptance in mathematics practice in the 20th century . The Zermelo set theory of 1907 is both the basis of the Zermelo-Fraenkel set theory (ZFC) and alternative axiom systems. ZFC results from the addition of Abraham Fraenkel's substitution axiom from 1921 and Zermelo's foundation axiom from 1930. The originally verbal set axioms by Zermelo-Fraenkel were later strictly formalized under the influence of Hilbert's program , which was supposed to ensure the consistency of fundamental axiom systems of mathematics. The first formalization (ZFC without foundation) by Thoralf Skolem from 1929 gave the impetus for modern predicate logic ZFC axiom systems . So far, no contradictions could be derived in ZFC. However, only the general set theory is demonstrably free of contradictions; according to Fraenkel, that is the ZFC set theory without the axiom of infinity , for which Zermelo 1930 gave a model. However, Hilbert's program could not be carried out for the complete Zermelo-Fraenkel set theory, since Gödel's incompleteness theorem also applies to them, so that their consistency cannot be proven within the Zermelo-Fraenkel set theory.

The consistency in relation to the Zermelo-Fraenkel set theory is also ensured for many extensions, generalizations and modifications. One of them is the set theory of John von Neumann from 1925, which is based on the functional concept instead of the set concept and not only includes sets but also real classes . It formed the starting point for the Neumann-Bernays-Gödel set theory , which ZFC generalizes for classes and uses a finite number of axioms, while ZFC needs axiom schemes. The Ackermann set theory of 1955, which tries to interpret Cantor's definition of sets in a precise and axiomatic manner, is even more general . Arnold Oberschelp embedded ZFC in a general axiomatic class logic in 1974 , so that his set theory allows a comfortable, syntactically correct representation with any class terms.

The well-known axiomatizations, which are not based on Cantor or Zermelo-Fraenkel, but on type theory, include the set theory of Willard Van Orman Quine (especially his New Foundations (NF) from 1937) and its extension Mathematical Logic (ML) from 1940.

Cantor's set axioms

It is noteworthy that it was Cantor who first presented a list of set theory axioms in 1898. He shared his rules for amount formation with the Cantor antinomies together Hilbert and Dedekind with letter. However, they were not published until 1932, so that they did not have any historical effect. Among other things, he formulated the following five rules for the formation of sets or finished sets (fert. M.):

  • “If you substitute in a fert. Set instead of the elements finished sets, the resulting multiplicity is a finished set. M. "
  • "The multiplicity of all partial sets of a finished set M is a finished set."
  • "Every sub-multiplicity of a set is a set".
  • "Any multitude of sets is, if the latter is broken down into their elements, also a set."
  • “That the 'countable' multiplicities {α ν } are complete sets seems to me to be an axiomatically safe proposition.” The index ν stands for finite cardinal numbers.

In the set theory that Cantor published, he did not mention such rules, but applied them tacitly. Occasionally, however, you will also find unproven sentences there, including the following:

  • "That it is always possible to bring any well-defined set into the form of a well-ordered set, [...]."

Cantor thought he could derive these rules from his definition of sets, but did not provide any evidence. Its rules are approximately equivalent to ZFC without foundation : Rule 1 corresponds to Fraenkel's axiom of substitution , rule 2 to the axiom of power sets , rule 3 to the axiom of separation , rule 4 to the axiom of union , rule 5 is equivalent to Zermelo's axiom of infinity ; the last sentence, the so - called well-ordered sentence, is equivalent to Zermelo's axiom of choice . Only the axiom of extensionality is missing .

literature

See also

Individual evidence

  1. ↑ Thoralf Skolem: About some basic questions of mathematics (1929) . In: Selected works in logic . Oslo 1970, p. 227-273 .
  2. ↑ Abraham Fraenkel: Axiomatic theory of ordered sets . In: Journal for pure and applied mathematics . tape 155 , 1926, pp. 129-158, especially p. 132 f .
  3. ↑ Wilhelm Ackermann: The consistency of general set theory , 1936, In: Mathematische Annalen. 114: 305-315 (1937).
  4. ↑ Ernst Zermelo: limit numbers and quantity ranges . In: Fundamenta Mathematicae . tape 16 , 1930, p. 29–47, especially p. 44 .
  5. ↑ John von Neumann: An axiomatization of set theory . In: Journal for pure and applied mathematics . tape 154 , 1925, pp. 219-240 .
  6. ^ Ulrich Felgner: The axiomatization of set theory , in: Ernst Zermelo: Collected Works I, Berlin Heidelberg 2010, p. 175: "However, it is remarkable that it was Cantor who in 1898 first represented a list of set-theoretic axioms. "
  7. ^ A b c Letter from Cantor to Hilbert dated August 10, 1898 in: Georg Cantor, Briefe, ed. H. Meschkowski and W. Nilson, Berlin, Heidelberg, New York 1999, p. 396.
  8. ↑ a b Letter from Cantor to Dedekind of August 3, 1899 in: Georg Cantor, Briefe, ed. H. Meschkowski and W. Nilson, Berlin, Heidelberg, New York 1999, p. 407.
  9. ↑ Georg Cantor: About infinite point manifolds , Article 5, in: Mathematische Annalen 21 (1883), p. 550; referred to there as the fundamental law of thought.