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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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.
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.