Preface -- Contents -- I. Introduction -- II. Infinity, Mathematics' Persistent Suitor -- 1. Incommensurable Lengths, Irrational Numbers -- 2. Newton and Leibniz -- 3. Go Forward, and Faith Will Come to You -- 4. Vibrating Strings -- 5. Infinity Spurned -- 6. Infinity Embraced -- III. Sets of Points -- 1. Infinite Sizes -- 2. Infinite Orders -- 3. Integration -- 4. Absolute vs. Transfinite -- 5. Paradoxes -- IV. What Are Sets? -- 1. Russell -- 2. Cantor -- 3. Appendix A: Letter from Cantor to Jourdain, 9 July 1904 -- 4. Appendix B: On an Elementary Question of Set Theory -- V. The Axiomatization of Set Theory -- 1. The Axiom of Choice -- 2. The Axiom of Replacement -- 3. Definiteness and Skolem's Paradox -- 4. Zermelo -- 5. Go Forward, and Faith Will Come to You -- VI. Knowing the Infinite -- 1. What Do We Know? -- 2. What Can We Know? -- 3. Getting from Here to There -- 4. Appendix -- VII. Leaps of Faith -- 1. Intuition -- 2. Physics -- 3. Modality -- 4. Second-Order Logic -- VIII. From Here to Infinity -- 1. Who Needs Self-Evidence? -- 2. Picturing the Infinite -- 3. The Finite Mathematics of Indefinitely Large Size -- 4. The Theory of Zillions -- IX. Extrapolations -- 1. Natural Models -- 2. Many Models -- 3. One Model or Many? Sets and Classes -- 4. Natural Axioms -- 5. Second Thoughts -- 6. Schematic and Generalizable Variables -- Bibliography -- Index.