Foundations of mathematics

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Chapter 1 Foundations of mathematics

Sets are the atoms from which most of modern mathematics is built, and the rules for combining them are carefully prescribed. These rules are the domain of logic and set theory, and we will develop these concepts in an informal — but still precise! — manner.

The basic ways of combining logical statements — conjunction, disjunction, and negation — are reflected in the set-theoretic operations of intersection, union, and formation of complements. Along with the conditional and biconditional, the rules for manipulating statements using these connectives is the domain of propositional logic. By introducing the universal and existential quantifiers, we complete our logical arsenal.

Using these ideas of logic and set theory, we are then able to describe our first number system — the natural numbers — and the idea of induction.