Type theory

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The type theory, developed by the mathematician Bertrand Russell form of set theory, with which he attempted, among other things, to eliminate the he discovered Russell's antinomy and other contradictions of the original or naive set theory.

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According to the theory of types, there are simple sets which can contain only original elements, but no quantities as elements; Quantities of the second type can only contain simple volumes, quantities of the third type etc. quantities have only quantities of the second type that is a higher one type as its elements. This system is thus based on a stratification of the quantity term, thereby avoiding Russell's antinomy, since the representation of the set of all sets not containing themselves is not simply for syntactic reasons, because the statements required for this purpose and are syntactically incorrect.

In a cumulative type theory any amount lower level or primal elements may be present in an amount which Russell's antinomy is also avoided.

The first simple version of type theory formulated Russell in 1903 in the appendix of his Principles of Mathematics, the second more complex version in 1908 in Mathematical Logic as based on the Theory of Types. The latter was the basis of the famous Principia Mathematica, which he published after almost ten years of preparation, together with Alfred North Whitehead in 1910. The simple type theory is since Ramsey also called Simplified Theory of Types ( or just STT); it is sufficient to avoid the logical contradictions such as Russell's antinomy, but not for solving the semantic contradiction as the Grelling -Nelson's antinomy; this only can its much more complex issue from 1908, the ramified theory of types ( RTT), the " branched " type theory.

This version of the theory of types sat down because of their complexity and limited performance not on a permanent basis. As a convenient and powerful proved in 1907, developed by Ernst Zermelo axiomatic set theory, the 1922 expanded and which is, in short, ZF, known as Zermelo -Fraenkel set theory and widely accepted as the basis of mathematics Abraham Fraenkel today. In addition, one has to ZF for a foundation axiom equivalent von Neumann hierarchy, which is a kind of stratification, but brings no other restrictions with it. The levels or ranks of quantities also correspond to a cumulative type theory.

A simplified variant of the theory of types designed Quine 1937 in his New Foundations.

Type theories play an important role in the theory of programming and programming languages ​​based on them, and computer-aided proof systems such as Coq and Agda (dependent types).

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