[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.
Thus Turing showed how one can associate logic with any constructive ordinal.
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