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We hope to accomplish four things by writing this book. Our rst goal is to o er a tutorial on the new mathematical ideas which underlie our research. In doing so we have tried to provide several entry points into the material, even at the cost of considerable redundancy. We hope that many of the ideas will be accessible to a well {trained undergraduate with a good background in mathematics and computer science. Second, parts of this book should serve as a manual for users of the Nuprl system (pronounced\new pearl"). As the system has grown so has the demand, both here at Cornell and at other institutions, for better documentation. We have tried to collect here all material relevant to the operation of the system. Third, we give an overview of our project for those interested in applications of the results and for those inclined to basic research in the area. Finally, we present new research which has arisen as we have worked on the Nuprl system. This system embodies contributions to the foundations of computer science and semantics, to automated reasoning and to system design, and it has shown promise as an intelligent system.

Authoring this book was a collective task; many individual e orts found their ways into these pages. Most chapters have several authors. That we were able to proceed in this fashion owes to the fact that we have assembled at Cornell a unique group of computer scientists who had worked for more than two years on the project.