Sets for Mathematics

The book introduces set theory as an algebra of mappings. This approach translates usual notions (sums, products, axiom of choice and others) to the language of mappings. The category S of abstract sets and mappings is defined using several axioms (S is a category; S has all finite limits and colimits; for any objects X and Y in S, there is a power X to Y; representation of truth values; S is Boolean; S is two-valued; axiom of choice). The category S of abstract sets and arbitrary mappings is a topos that is two-valued with an infinite object and the axiom of choice. This abstract approach includes all known situations in one simple frame. The material presented in the book is illustrated with many useful exercises. The book is suitable for advanced undergraduates or those beginning graduate studies. It gives a well-founded basis for the study of mathematics. The book consists of 10 chapters devoted to abstract sets and mappings; sums, monomorphisms, finite inverse limits; colimits, epimorphisms and the axiom of choice; mapping sets and exponentials; consequences and uses of exponentials; power sets; variable sets and models of additional variation; together with three appendices (logic, maximal principles, definitions, etc.). Many diagrams illustrate the book. It is an excellent book for those mathematicians who wish to study foundations of mathematics in an axiomatic form based on an algebraic approach.

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