mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, ordinal logic is a logic associated with an

ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...

by recursively adding elements to a sequence of previous logics.Solomon Feferman, ''Turing in the Land of O(z)'' in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 page 111''Concise Routledge encyclopedia of philosophy'' 2000 page 647 The concept was introduced in 1938 by

Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...

in his PhD dissertation at Princeton in view of

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phil ...

.Alan Turing, ''Systems of Logic Based on Ordinals'' Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–22

/ref> While Gödel showed that every logic system suffers from some form of incompleteness, Turing focused on a method so that a complete system of logic may be constructed from a given system of logic. By repeating the process a sequence L1, L2, … of logic is obtained, each more complete than the previous one. A logic L can then be constructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc. Thus Turing showed how one can associate logic with any constructive ordinal.

References

Mathematical logic Systems of formal logic Ordinal numbers {{mathlogic-stub