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At the heart of the justification for the reasoning used in modern mathematics lies the completeness theorem for predicate calculus. This unique textbook covers two entirely different ways of looking at such reasoning. Topics include:

• the representation of mathematical statements by formulas in a formal language;
• the interpretation of formulas as true or false in a mathematical structure;
• logical consequence of one formula from others;
• formal proof;
• the soundness and completeness theorems connecting logical consequence and formal proof;...

This volume is designed for undergraduate students of set theory. It covers the classic work of Georg Cantor and Richard Dedekind at the end of the 19th century, and their immediate successors, viewed from a modern perspective. The book includes: the definition of the real numbers in terms of rational numbers and ultimately in terms of natural numbers; defining natural numbers in terms of sets; the potential paradoxes in set theory; the Zermelo-Fraenkel axioms for set theory; the axiom of choice; the arithmetic of ordered sets; Cantor's two sorts of transfinite number, cardinals and ordinals;...