Ordinal logic

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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 system of logic suffers from some form of incompleteness, Turing focused on a method so that from a given system of logic a more complete system may be constructed. By repeating the process a sequence L1, L2, … of logics 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 a logic with any constructive ordinal.[3]


  1. ^ a b Solomon Feferman, Turing in the Land of O(z) in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 ISBN 3-211-82637-8 page 111
  2. ^ Concise Routledge encyclopedia of philosophy 2000 ISBN 0-415-22364-4 page 647
  3. ^ a b Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[1]