Realizability: A Historical Essay

نویسنده

Jaap van Oosten

چکیده

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 re ects very much my own personal taste biases and prejudices Realizability has over the past 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 di erent 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 su ering 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 classical 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 meta mathematics of intuitionistic arithmetical theories and topos theoretic devel opments

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Realizability: An Historical Essay

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