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

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IV.23 Logic and Model Theory

David Marker

1 Languages and Theories

Mathematical logic is the study of formal languages that are used to describe mathematical structures and what these can tell us about the structures themselves. We can learn a lot about a formal language by investigating which of its sentences are true for the structure it describes, and we can learn a lot about the structure by investigating the subsets of it that can be defined using the language. In this article, we shall see several examples of languages and the structures that they are used to describe. We shall also see instances of the remarkable phenomenon that theorems in logic can sometimes be used to prove “purely mathematical” results that seem to have nothing to ...

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