Heyting

Brouwer's bar theorem is crucial to intuitionistic analysis; for a detailed explanation of the notions involved and of Brouwer's proof, see Heyting 1956 (Ch. 3), Parsons 1967, and van Atten 2004b

The Development of Intuitionistic Logic van Atten, Mark 2009

Heyting himself spoke simply of the “interpretation”

The Development of Intuitionistic Logic van Atten, Mark 2009

The standard explanation of intuitionistic logic today is the BHK-Interpretation (for “Brouwer, Heyting, Kolmogorov”) or Proof Interpretation as given by Troelstra and Van Dalen in

The Development of Intuitionistic Logic van Atten, Mark 2009

The name “Proof Interpretation” for the explanation that Heyting published in the 1930s and later seems to have made its first appearance in print only in 1973, in papers by Van Dalen and Kleene, presented at the same conference (van Dalen 1973a, Kleene 1973).

The Development of Intuitionistic Logic van Atten, Mark 2009

We will therefore begin our account of the historical development of intuitionistic logic with Brouwer's ideas, and then show how, via Heyting and others, the modern Proof Interpretation was arrived at.

The Development of Intuitionistic Logic van Atten, Mark 2009

(Heyting in 1956 chose to define implication in this stronger sense; see section 5.4 below.)

The Development of Intuitionistic Logic van Atten, Mark 2009

A variety of interpretations for intuitionistic logic have been extended to intuitionistic and constructive set theories, such as realisability, Kripke models and Heyting-valued semantics.

Set Theory: Constructive and Intuitionistic ZF Crosilla, Laura 2009

The category Heyt with objects Heyting algebras and

Category Theory Marquis, Jean-Pierre 2007

We now consider Gödel 1933e, in which Gödel showed, in effect, that intuitionistic or Heyting arithmetic is only apparently weaker than classical first-order arithmetic.

Kurt Gödel Kennedy, Juliette 2007

(H is intuitionistic propositional logic, after Heyting.)

Kurt Gödel Kennedy, Juliette 2007