Realizability A Historical Essay Jaap van Oosten Department of Mathematics POBox TA Utrecht The Netherlands jvoostenmathuunl January Dedicated to Anne S Tro elstra at his th Birthday Intro duction The purp ose of this short pap er is to sketch the development of a few basic topics in the history of Realizability The numb er of topics is quite limited and reects very muchmyown p ersonal taste biases and prejudices Realizability has over the past years develop ed into a sub ject of such dimensions that a comprehensiveoverview would require a fat b o ok Mayb e someone some day ought to write such a b o ok But it will not b e 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 dierent areas of Logic Mathematics and Computer Science and this proliferation shows no signs of diminishing in our days Likeavenomous carci noma Realizability stretches out its tentacles to ever more remote elds Linear Logic Complexity Theory and Rewrite Theory have already b een infected The theory of Subrecursive Hierarchies to o Everything connected to the calculus is heavily engaged Pro of Theory is suering Intuitionism is dead Just to name a few Did you think that at least the realm of classical logic would b e safe Recently Krivine came up with a Realizabilityinterpretation for ZF set theory Confronted with this mess I have acted like the classical imp ostor who walked into the hospital claiming to b e a surgeon and is now wielding the knives in the op erating theatre I to ok the nearest scalp el at hand and cut out everything that wouldnt t into either one of mytwo ma jor streams meta mathematics of intuitionistic arithmetical theories and top ostheoretic devel opments Research supp orted by PIONIER NWO the Netherlands 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 ap ologies just reverb erate in the vast emptiness I have created Therefore lets get physical and say something concrete ab out what is in this pap er According to me there are three landmark publications in Realizability These are Kleenes original pap er On the Interpretation of Intuitionistic Num b er Theory Tro elstras Metamathematical Investigations from Hylands The EectiveTop os from Of these three b oth and initiated a whole new strand of research I have therefore decided that the material I wished to present naturally divides into two periods viz and This is not to say that suddenly there were after 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 of these matters still need and deservetobeinvestigated and need all the elegance and exp ository skills we can muster no radically new vistas have emerged from this research Therefore in my accountofthe second p erio d I have concentrated on what I regard as more innovative research The second item in my list is of a dierent kind This monumental work brought together all existing results many of whichwere due to its author and ordered them in suchaway that the diligent student could see at once the similarities b etween them It charted the territory and in this wayachieved something of conceptual value the notion that all these systems interpretations and axiomatizations were manifestations of a pattern that they had in common What exactly this pattern is we still dont know But it is my feeling that the categorical analyses of later years owe a lot to this work It made when it app eared a daunting impression on some p eople And certainly it did so on me when I was his student But now I exp erience a sensation of dry austere b eauty in its relentless pursuit of order And let us not forget it set new standards of presentation and notation For although Kleenes rst pap er is a gem of readability regrettably Kleene later adopted a style of writing whichwas so cluttered with notation that it takes a strong man to ght himself through Ihave therefore decided to dedicate this pap er to Anne Tro elstra mymentor who has contributed so much to the sub ject matter in gratitude Acknowledgements I am grateful to Joan Moschovakis who had a careful lo ok at a preliminary version and drew my attention to a few embarrassing mistakes Id also like to thank Lars Birkedal Martin Hyland Pino Rosolini and Dana Scott for discussions and for providing lastminute bibliographical and background information Let the disapp ointed reader b e solaced by the availabilityofan excel lent pro oftheoretical survey on Realizability The rst years The origin of Realizability In his overview pap er Realizability a retrosp ectivesurvey Stephen Cole Kleene recounts how his idea for numerical realizability develop ed He wished to give some precise meaning to the intuition that there should b e a connection b etween Intuitionism and the theory of recursive functions b oth the ories stressing the imp ortance of extracting information eectively He started to think ab out this in In order to appreciate the originality of his thinking one should recall that the formal system of intuitionistic arithmetic HA did not exist at the time Well There is a system closely resembling HA in Godels pap er Kleene ap p ears to have b een at least initially unaware of this for although his pap er gives the reference the retrosp ectivesurvey stresses that Heyting Arithmetic do es not o ccur as a subsystem readily separated out from Heytings full sys tem of intuitionistic mathematics and quotes Kleenes own formalismwhich later app eared in as the thing he had in mind As an example of a precise connection b etween Intuitionism and the theory of recursive functions Kleene starts by conjecturing a weak form of Churchs Rule if a closed formula of the form xyx y isprovable in intuitionistic numb er theory then there must b e a general recursive function F such that for all n the formula n F n is true One arrives at this conjecture by unravelling the meaning that such a statementmust have for an intuitionist Conjecturing this at a time when Intuitionism was still clouded by Brouwers mysticism the formal system in question hardly established and the contentof the conjecture blatantly false for Peano Arithmetic was imaginative indeed But this was still far away from the actual development of Realizability Often one encounters the opinion that Realizabilitywas inspired by the so called BrouwerHeytingKolmogorovinterpretation a mantra which rather than b eing an interpretation is itself in need of one This was not the case Kleene starts by quoting Hilb ert and Bernays They in their Grundlagen der Mathematik explain the nitist p osition in Mathematics The relevant passage is the one ab out existential statements as incomplete communications which since it is philosophy can only b e appropriately understo o d in the original German Ein Existenzsatz ub er Ziern also ein Satz von der Form es gibt eine Zier n von der Eigenschaft An ist nit aufzufassen als ein Partialurteil dh als eine unvollstandige Mitteilung einer genauer b estimmten Aussage welche entweder in der direkten Angab e einer Zier von der Eigenschaft An o der der Angab e eines Verfahrens zur Gewinnung einer solchen Zier b esteht For some biographical details on Kleene and a p ersonal appreciation see the obituary by his friend Saunders Mac Lane An existential statement ab out numb ers ie a statement of the form there exists a Kleene then asks Can we generalize this idea to think of al l except trivially the simplest intuitionistic statements as incomplete communications He outlines in which sense every logical sentence is incomplete and what would constitute its completion For the implication case Kleene inter estingly says that rst he tried an inductive clause inspired by Heytings pro ofinterpretation but that it didnt work and so Heytings pro of interpretation failed to help me to my goal Since Kleene do esnt reveal what this rst try was we are free to conjecture It is just conceivable that he tried a realizer for A B is a partial recursive function which sends pro ofs of A to pro ofs of B Kleenes realizabilitywas at least conceptually a ma jor advance Its achieve ment is not so much a philosophical explanation of the intuitionistic connectives Tro elstra p says it cannot b e said to maketheintended meaning of the logical op erators more precise As a philosophical reduction of the inter pretation of the logical op erators it is also mo derately successful eg negative formulae are essentially interpreted by themselves In fact Kleene admits this explicitly in his pap er On the other hand by providing an interpretation which can b e read and checked by the classical mathematician he did put for ward an interpretation of the intuitionistic connectives in terms of the classical ones this in contrast to the socalled BHK or pro of interpretation which interprets the intuitionistic connectives in terms of themselves More imp ortantly realizability as it is designed to handle information ab out formulas rather than pro ofs already hints at the role Intuitionism would come to play in theoretical Computer Science some years later it foreshadows the view of intuitionistic formulas as datatypes and intuitionistic logic as the logic of information But the scop e of realizability is wider than just interpreting the logic Re alizability also provides mo dels for theories which are