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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,
The inadequacy of information regarding fishweirs also impinges on reconstructions of prehistoric subsistence and settlement patterns. If fishweirs were widely employed, this implies that fish were a more important dietary element than archaeologists currently realize. Similarly, the use of weirs may have affected settlement patterns because of the time and seasonal constraints necessitated by their construction, repair, and utilization.