Ordinal logic

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 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.^[3]

References

↑ ^1.0 ^1.1 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
↑ Concise Routledge encyclopedia of philosophy 2000 ISBN 0-415-22364-4 page 647
↑ ^3.0 ^3.1 Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[1]

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[feferman-1] 1.0 ^1.1 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

[alan-3] 3.0 ^3.1 Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[1]

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