How can the infinite, a subject so remote from our finite experience, be an everyday working tool for the working mathematician? Blending history, philosophy, mathematics and logic, Shaughan Lavine answers this question with clarity. An account of the origins of the modern mathematical theory of the infinite, his book is also a defense against the attacks and misconceptions that have dogged this theory since its introduction in the late 19th century. With his development of set theory in the 1880s, Georg Cantor introduced the infinite into mathematics. But his theory, both critics and supporters have charged, was subject to paradoxes proceeding from Cantor's "naive intuitions", and this verdict has had an enormous impact on the philosophy of mathematics. Lavine effectively reverses this charge by showing that set theory is in fact an excellent example of the posititve and necessary role of intuition in mathematics. His history, moving from Greek geometry through the development of calculus to the evolution of set theory, ultimately leads to the crux of the issue: the source of our intuitions concerning the infinite. Along the way, he offers a careful and critical discussion of differing views across the philosophical spectrum. Making use of the mathematical work of Jan Mycielski, formerly accessible only to logicians, Lavine demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge. He shows that the source of our intuitions concerning Cantor's infinite, as a matter of historical and psychological fact, is extrapolation from ordinary experience of the indefinitely large.

Table of Contents:
Part 1 Introduction. Part 2 Infinity, mathematics' persistent suitor: incommensurable lengths, irrational numbers; Newton and Leibniz; go forward, and faith will come to you; vibrating strings; infinity spurned; infinity embraced. Part 3 Sets of points: infinite sizes; infinite orders; integration; absolute versus transfinite; paradoxes. Part 4 What are sets?: Russell; Cantor; appendix A - letter from Cantor to Jourdain, 9 July 1904; appendix B - on an elementary question of set theory. Part 5 The axiomatization of set theory: the axiom of choice; the axiom of replacement; definiteness and Skolem's paradox; Zermelo; go forward, and faith will come to you. Part 6 Knowing the infinite: what do we know?; what can we know?; getting from here to there. Part 7 Leaps of faith: intuition; physics; modality; second-order logic. Part 8 From here to infinity: who needs self-evidence?; picturing the infinite; the finite mathematics of indefinitely large size; the theory of zillions. Part 9 Extrapolations: natural models; many models; one model or many? sets and classes; natural axioms; second thoughts; schematic and generalizable variables.