Incomputability Emergent, and Higher Type Computation

Typing of information played an historical role in bringing consistency to formulations of set theory and to the foundations of mathematics. Underlying this was the augmentation of language and logical structure with a respect for constructive principles and the corresponding infrastructure of an informational universe. This has important consequences for how we view the computational character of science, the humanities and human creativity. The aim of this article is to make more explicit the anticipations and intuitions of early pioneers such as Alan Turing in informing and making relevant the computability theoretic underpinnings of today’s understanding of this. We hope to make clearer the relationship between the typing of information—a framework basic to all of Turing’s work—and the computability theoretic character of emergent structure in the real universe. The informational terrain arising is one with comprehensive computational structure, but with theoretical boundaries to those areas accessible to effective exploration in an everyday setting.