I am reading Turing's 1939 paper on ordinal logic ("Systems of Logic Based on Ordinals", A. M. Turing, Proc. London Math. Soc. ser. 2, 45 (1939), #1, 161-228, DOI: 10.1112/plms/s2-45.1.161.) Immediately after talking about successively extending axiomatic systems using transfinite iteration of the reflection principle, he says, on p. 197: "Another ordinal logic of this type has in effect been introduced by Church". Does anyone know what this is referring to? Turing gives two references in a footnote, but the first gives no details and the second is mimeographed notes which I don't have.

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    $\begingroup$ For those who want to take a look: Turing writes that the ordinal logic is in outline contained on pages 279-280 of Church's A Proof of Freedom from Contradiction, PNAS 1935 21 (5) 275-281, available here. $\endgroup$
    – t.b.
    Jan 5, 2012 at 7:07

2 Answers 2


Turing was Church's PhD student when he developed ordinal logic. Church apparently suggested the idea of iterated consistency statements as a way to escape incompleteness. This Feferman paper talks a bit about it.

Turing [...] decided to stay the extra year and do a Ph.D. under Church. Proposed as a thesis topic was the idea of ordinal logics that had been broached in Church’s course as a way to "escape" Gödel's incompleteness theorems.


I can only refer you to Solomon Feferman's paper Turing in the Land of 0(z) where the author chiefly discusses Turing's idea of his third paper which you mentioned. (Wikipedia does not contain that much of information on ordinal logic.)

Edit: Google books' copy of Feferman's paper can be accessed here.


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