Mathematical Constructivism - Intuitionism, Constructivism, Ultrafinitism, Intuitionistic Logic, Heyting Algebra, Brouwer-Hilbert Controversy, Criti (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 39. Chapters: Intuitionism, Constructivism, Ultrafinitism, Intuitionistic logic, Heyting algebra, Brouwer-Hilbert controversy, Criticism of non-standard analysis, Modulus of continuity, Intuitionistic type theory, Constructive set theory, Brouwer-Heyting-Kolmogorov interpretation, Constructive analysis, Primitive recursive arithmetic, Constructive proof, Choice sequence, Markov's principle, Realizability, Church's thesis, Harrop formula, Apartness relation, Diaconescu's theorem, Inhabited set, Friedman translation, Indecomposability, Disjunction and existence properties, Pseudo-order, Axiom schema of predicative separation, Heyting arithmetic, Minimal logic, Modulus of convergence, Bar induction, Computable analysis, Computable model theory, Subcountability, Heyting field. Excerpt: In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a b of implication such that (a b) a b, and moreover a b is the greatest such in the sense that if c a b then c a b. From a logical standpoint, A B is by this definition the weakest proposition for which modus ponens, the inference rule A B, A B, is sound. Equivalently a Heyting algebra is a residuated lattice whose monoid operation a b is a b; yet another definition is as a posetal cartesian closed category with all finite sums. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. As lattices, Heyting algebras can be shown to be distributive. Every Boolean algebra is a Heyting algebra when a b is defined as usual as -a b, as is every complete distributive lattice when a b is taken to be the supremum of the set of all c for which a c b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra....