The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. . . - David Hilbert (1862-1943) Infinity is a fathomless gulf, There is a story attributed to David Hilbert, the preeminent mathe­ into which all things matician whose quotation appears above. A man walked into a vanish. hotel late one night and asked for a room. "Sorry, we don't have o Marcus Aurelius (121- 180), Roman Emperor any more vacancies," replied the owner, "but let's see, perhaps and philosopher I can find you a room after alL" Leaving his desk, the owner reluctantly awakened his guests and asked them to change their rooms: the occupant of room #1 would move to room #2, the occupant of room #2 would move to room #3, and so on until each occupant had moved one room over. To the utter astonish­ ment of our latecomer, room #1 suddenly became vacated, and he happily moved in and settled down for the night. But a numbing thought kept him from sleep: How could it be that by merely moving the occupants from one room to another, the first room had become vacated? (Remember, all of the rooms were occupied when he arrived.

I. Mathematical Infinity.- 1. First Steps to Infinity.- 2. Towards Legitimation.- 3. Convergence and Limit.- 4. The Fascination of Infinite Series.- 5. The Geometric Series.- 6. More about Infinite Series.- 7. Interlude: An Excursion into the Number Concept.- 8. The Discovery of Irrational Numbers.- 9. Cantor's New Look at the Infinite.- 10. Beyond Infinity.- II. Geometric Infinity.- 11. Some Functions and Their Graphs.- 12. Inversion in a Circle.- 13. Geographic Maps and Infinity.- 14. Tiling the Plane.- 15. A New Look at Geometry.- 16. The Vain Search for Absolute Truth.- III. Aesthetic Infinity.- 17. Rejoice the Infinite!.- 18. The Möbius Strip.- 19. The Magic World of Mirrors.- 20. Horror Vacui, Amor Infiniti.- 21. Maurits C. Escher-Master of the Infinite.- 22. The Modern Kabbalists.- IV. Cosmological Infinity.- 23. The Ancient World.- 24. The New Cosmology.- 25. The Horizons Are Receding.- 26. A Paradox and Its Aftermath.- 27. The Expanding Universe.- 28. The Modern Atomists.- 29. Which Way from Here?.- Epilogue.