# Ordinal logic

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In [mathematics](/source/Mathematics), **ordinal logic** is a logic associated with an [ordinal number](/source/Ordinal_number) by recursively adding elements to a sequence of previous logics.[1][2] The concept was introduced in 1938 by [Alan Turing](/source/Alan_Turing) in [his PhD dissertation](/source/Systems_of_Logic_Based_on_Ordinals) at [Princeton](/source/Princeton) in view of [Gödel's incompleteness theorems](/source/G%C3%B6del's_incompleteness_theorems).[3][1]

While [Gödel](/source/G%C3%B6del) showed that every [recursively enumerable](/source/Computably_enumerable_set) [axiomatic system](/source/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](/source/Constructive_ordinal).[3]

## References

1. ^ [***a***](#cite_ref-feferman_1-0) [***b***](#cite_ref-feferman_1-1) [Solomon Feferman](/source/Solomon_Feferman), *Turing in the Land of O(z)* in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 [ISBN](/source/ISBN_(identifier)) [3-211-82637-8](https://en.wikipedia.org/wiki/Special:BookSources/3-211-82637-8) page 111

1. **[^](#cite_ref-2)** *Concise Routledge encyclopedia of philosophy* 2000 [ISBN](/source/ISBN_(identifier)) [0-415-22364-4](https://en.wikipedia.org/wiki/Special:BookSources/0-415-22364-4) page 647

1. ^ [***a***](#cite_ref-alan_3-0) [***b***](#cite_ref-alan_3-1) Alan Turing, *Systems of Logic Based on Ordinals* Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[\[1\]](https://web.archive.org/web/20141119022238/http://plms.oxfordjournals.org/content/s2-45/1/161.extract)

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