Fixed Points, Induction, and Coinduction in Order Theory, Set Theory, Type Theory, Category Theory, and Logic: A Concise Summary
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In this note we present the formulation of the induction and co-induction principles and related notions, such as fixed points, using the language and conventions of each of order theory, set theory, (first-order) logic, category theory, and the theory of types in (object-oriented and functional) programming languages, for the purpose of examining some of the similarities and dissimilarities between these six mathematical subdisciplines. As a side-benefit that is of relevance to programming language researchers in particular, our comparison demonstrates one of the fundamental differences between structural typing (dominant in FP) and nominal typing (dominant in mainstream OOP).
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