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Incomputability after Alan Turing

Incomputability as a mathematical notion arose from work of Alan Turing and Alonzo Church in the 1930s. Like Turing himself, it attracted less attention than it deserved beyond the confines of mathematics. Today our experiences in computer science, physics, biology, artificial intelligence, economics and the humanities point to the importance of the notion for understanding the world around us. This article takes a Turing centenary look at how the interface between the computable world and the incomputable formed a central theme in Turing's work - from the early establishment of the standard model of the stored program computer, to the late work on emergence of form in nature, and the first approaches to understanding and simulating human intelligence. Turing's thinking was remarkably prescient, and his legacy still impacts on much of our work. The incomputable may turn out to be a specially important part of the legacy, with consequences that Alan Turing himself could not have envisaged.