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Kleene himself never stated that Turing had made a mistake in his paper, important in its own right for helping to establish the unsolvability of problems in group theoretic computations, although corrections to Turing's paper were also made later by Boone who originally pointed out "points in the proof require clarification, which can be given"  and Turing's only phd student, Robin Gandy. That Kleene doesn't mention this mistake in the body of his textbook where his presents his work on Turing machines but buried the fact he was correcting Alan Turing in the appendix was appreciated by Turing himself can be surmised from the ending of Turing's last publication "Solvable and Unsolvable Problems" which ends not with a bibliography but the words,
ERIC BERNE was an objective scientist, a highly trained physician and psychiatrist who rebelled against the vague, experiential subjective world of the current psychoanalytic establishment and their complicated language. To their charges to him of “intellectualizing,” he replied, “We respect our patients intelligence.” They would say to him “Yours is oversimplified,” and to that he retorted “Yours is overcomplicated.” At his home he gathered a professional group including many doctors; Goulding, Harris, Everts, Karpman, Dusay, Steiner, Kupfer, and others, and welcomed all disciplines equally, to help set down a new game-changing set of strict standards of greater objectivity in psychiatric thought than had ever been known before. At the core of his theory was the observable social transaction, a transactional reality in objective facts that could be “photographed and tape recorded,” as opposed to imaginary transferences and an invisible Super Ego, Ego, and Id that could not be “photographed and tape recorded.” He had a high standard of rigorous objective thinking about what real people do in the real world, not in the protected subjective world of the analysts couch.
This could all be a huge misdirect, sure, but it’s definitely starting to look like everyone’s assumptions about Jon Snow were very, very accurate.