 Church was acutely conscious of the effect on the question of foundations of Gödel’s incompleteness theorems. His early attitude was that they were “unpleasant” results to be circumvented, if at all possible (1934). His work in this direction is related to Alan Turing’s work on “ordinal logics” in his Princeton doctoral thesis completed under Church’s direction (see Turing’s [1939] paper and Feferman [1995]). Turing’s idea, encouraged by Church, was to extend an incomplete system by adding to it the sentences unprovable in it and iterating this operation into the transfinite. However, the iteration must be effective and involve only ordinals that are constructive in the sense defined and studied earlier by Church (1938) and Kleene (1938). This idea is still being investigated today.