Foundations 4: Logic and Partially Ordered Sets

In this series we develop an understanding of the modern foundations of pure mathematics, starting from first principles. We start with intuitive ideas about set theory, and introduce notions from category theory, logic and type theory, until we are in a position to understand dependent type theory, and in particular, homotopy type theory, which promises to replace set theory as the foundation of modern mathematics. We also take an interest in computer science, and how to write computer programming languages to formalize mathematics. In this video we introduce classical logic, using functional representations of and or, not and implies, and then we consider posets of subsets, introduce Boolean algebras, and eventually discuss Heyting Algebras as bicartesian closed categories, and discuss how our universal constructions of categorical products and coproducts give us ideas like AND and OR in intuitionistic logic. In this way we start to understand computational trinitarianism (which relates to the Curry-Howard-Lambek correspondence, and the Brouwer-Heyting-Kolmogorov interpretation).