In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics.[1] [2] The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.[3] [1]
While Gödel showed that every recursively enumerable axiomatic system that can interpret basic arithmetic 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.[3]