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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.22 Set Theory

Joan Bagaria

1 Introduction

Among all mathematical disciplines, set theory occupies a special place because it plays two very different roles at the same time: on the one hand, it is an area of mathematics devoted to the study of abstract sets and their properties; on the other, it provides mathematics with its foundation. This second aspect of set theory gives it philosophical as well as mathematical significance. We shall discuss both aspects of the subject in this article.

2 The Theory of Transfinite Numbers

Set theory began with the work of CANTOR [VI.54]. In 1874 he proved that there are more real numbers than there are algebraic ones, thus showing that infinite sets can be of different sizes. This also provided a new ...

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