Mathematician’s World

We start by explaining which objects are studied in mathematics. We follow the standard approach in which these objects are mathematical structures. We discuss the crucial role of sets in the foundations of mathematics and briefly sketch how they are used to define some basic arithmetical structures. In this introductory chapter we state the basic axioms about sets informally as principles. We present the most important antinomies of intuitive set theory. We discuss the axiomatic method, which was discovered in antiquity and is the basis of all contemporary mathematical theories. Finally, we explain why abstract concepts are useful, even if we only need to solve a problem stated in elementary terms.

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