https://speakerdeck.com/jonsterling 2016-08-18 07:33:57 -0400 The Church-Turing Thesis is a scientific hypothesis as to the nature of effective computation at lower type; the nature of higher type computability, however, is far from settled and admits many possible models. In the 1920s, L.E.J. Brouwer (the father of Intuitionism) gave a mysterious "proof" of an induction principle for infinite trees called the Bar Theorem, which has many consequences, including a constructive contrapositive to König's Lemma (the Fan Theorem). We situate the Bar Theorem more generally as "Brouwer's Thesis", a scientific claim about the nature of higher-type computability which may be contrasted with competing explanations, including recursive realizability and Turing machines. Then, we demonstrate using a technique called Effectful Forcing that Brouwer's Bar Theorem is constructively valid for any bar which can be written in Gödel's System T of primitive recursive arithmetic of finite type, a very simple programming language. The Church-Turing Thesis is a scientific hypothesis as to the nature of effective computation at lower type; the nature of higher type computability, however, is far from settled and admits many possible models. In the 1920s, L.E.J. Brouwer (the father of Intuitionism) gave a mysterious "proof" of an induction principle for infinite trees called the Bar Theorem, which has many consequences, including a constructive contrapositive to König's Lemma (the Fan Theorem). We situate the Bar Theorem more generally as "Brouwer's Thesis", a scientific claim about the nature of higher-type computability which may be contrasted with competing explanations, including recursive realizability and Turing machines. Then, we demonstrate using a technique called Effectful Forcing that Brouwer's Bar Theorem is constructively valid for any bar which can be written in Gödel's System T of primitive recursive arithmetic of finite type, a very simple programming language. Wed, 17 Aug 2016 00:00:00 -0400 https://speakerdeck.com/jonsterling/primitive-recursive-bars-are-inductive https://speakerdeck.com/jonsterling/primitive-recursive-bars-are-inductive