Turing's work was in algorithms, models of computation, and cryptanalysis. All that is considered part of Theory A, see e.g. http://blog.computationalcomplexity.org/2003/03/theory-and-t....
His doctoral advisor was Church, his work based on Gödel, his thesis named Systems of Logic Based on Ordinals.
Comments like these show the perils of treating mathematics like a cookbook. It's a very limiting mindset.