Realizability: An Historical Essay

Introduction The purpose of this short paper is to sketch the development of a few basic topics in the history of Realizability. The number of topics is quite limited and reeects very much my own personal taste, prejudices and area of competence. Realizability has, over the past 60 years, developed into a subject of such dimensions that a comprehensive overview would require a fat book. Maybe someone, some day ought to write such a book. But it will not be easy. Quite apart from the huge amount of literature to cover, there is the task of creating unity where there is none. For Realizability has many faces, each of them turned towards diierent areas of Logic, Mathematics and Computer Science, and this proliferation shows no signs of diminishing in our days. Like a venomous carci-noma, Realizability stretches out its tentacles to ever more remote elds: Linear Logic, Complexity Theory and Rewrite Theory have already been infected. The theory of Subrecursive Hierarchies too. Everything connected to the-calculus is heavily engaged. Proof Theory is suuering. Intuitionism is dead. Just to name a few! Did you think, that at least the realm of classical logic would be safe? Recently, Krivine came up with a Realizability interpretation for ZF set theory! Confronted with this mess, I have acted like the typical impostor who walked into the hospital claiming to be a surgeon, and is now wielding the knives in the operating theatre: I took the nearest scalpel at hand and cut out everything that wouldn't t into either one of my two major streams: metamathematics of intuitionistic arithmetical theories, and topos-theoretic developments. 1 Needless to say, there is no question of even starting to list what I omitted{ sometimes to my great regret, although I realize that such hollow apologies just reverberate in the vast emptiness I have created 1. Therefore, let's get physical and say something concrete about what is in this paper. According to me, there are three landmark publications in Realizability. Of these three, both 1) and 3) initiated a whole new strand of research. I have therefore decided that the material I wished to present, naturally divides into two periods, viz. 1940{1980 and 1980{2000. This is not to say that suddenly there were, after 1980, no more purely syntactical presentations of Realizabilities (quite on the contrary, thanks to Computer Science syntax is back!), but I do feel that although many …