Set theory#Axiomatic set theory
As axiomatic set theory is any axiomatization of set theory, which avoids the known antinomies of naive set theory. The most common axiomatization in modern mathematics is the Zermelo -Fraenkel set theory with the axiom of choice ( ZFC ). In addition, in category theory and certain parts of the Universe algebra axiom is often assumed.
History and characteristics
First axiomatization of set theory were tried before the discovery of quantity antinomies, namely in 1889 by Giuseppe Peano in 1893 by Gottlob Frege, both built the arithmetic on a volume or calculus classes. But as they both calculi - because of axioms that allow unlimited amount of education - proven to be inconsistent, one expects them to naive set theory. So sub axiomatic set theory is meant only such axiomatizations that seek to avoid restrictive amount by formation of these contradictions of naive set theory.
To avoid contradictions Bertrand Russell suggested a gradual build-up of set theory and developed his theory of types 1903-1908, which also served as the basis of Principia Mathematica 1910. In it has a lot always a higher type than their elements. Statements such as " this set contains itself as an element ", with the Russell's antinomy is formed, can not formulate in this theory. Thus, the type theory tries to solve through a restricted syntax of permissible class statements the problems. She has at Russell himself no axiomatic form, but was removed later to a relatively complicated axiomatic theory. Their consistency was demonstrated by Paul Lorenzen. The consistency of the building on the theory of types Principia Mathematica is not provable because of Gödel's incompleteness theorem. The type theory of Principia Mathematica was instrumental in the logic for a long time, but could in mathematics practice not prevail, both because of their complexity and on the other hand, because of their inadequacy. It namely not enough to justify Cantor's set theory and mathematics, as their linguistic means are too weak.
In math practice sat rather in the 20th century by gradually initiated by Ernst Zermelo form of axiomatic set theory. The Zermelo set theory of 1907 is both the basis of the Zermelo -Fraenkel set theory ( ZFC ) and alternative axiom systems. ZFC is obtained by addition of Abraham Fraenkel substitution axiom of 1921 and Zermelo foundation axiom of 1930. Originally verbal amount axioms of Zermelo - Fraenkel were strictly formalized under the influence of Hilbert's program, which should ensure the consistency of fundamental axiom systems of mathematics later. The first formalization ( ZFC without foundation ) by Thoralf Skolem 's 1929 gave impetus to modern predicate logic ZFC axiom systems. In ZFC been no contradiction could be derived. Demonstrably free of contradiction is just the general set theory, which is after the Fraenkel set theory ZFC without axiom of infinity, ie the set theory with finite sets; for she gave Zermelo 1930 model. For the complete Zermelo -Fraenkel set theory to Hilbert's program but did not perform as Gödel's incompleteness theorem applies to them, so their consistency is unprovable within the Zermelo -Fraenkel set theory.
The consistency relative to Zermelo -Fraenkel set theory is also secured for many extensions, generalizations and modifications. Among them is the set theory by John von Neumann in 1925, based on the concept of function rather than on the concept of a set and not only quantity but also includes proper classes. It was the starting point for the Neumann - Bernays - Gödel set theory, the ZFC for generalized classes and with a finite number of axioms manages, while ZFC axiom schemata needed. More generally, the Ackermann set theory from 1955 that attempts to interpret precise axiomatic Cantor set definition. Arnold Oberschelp 1974 ZFC embedded in a general axiomatic logic class, so his set theory allows convenient syntactically correct representation of any class terms.
Among the known axiomatizations that are not based on Zermelo - Fraenkel or Cantor, but to the type theory, set theory is one of Willard Van Orman Quine, in particular the New Foundations ( NF) from the year 1937.