TITLE: Turing's 1939 paper on ordinal logic
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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.