N. IA Vilenkin (1920–1991)

Indeholder "Foreword", "Preface", "1. Some Extraordinary Properties of Infinite Sets", " The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet", " From the Author", "2. Sets and Operations on Sets", " What Do We Mean by a Set?", " How We Specify a Set", " To Shave or Not to Shave?", " The Empty Set", " The Theory of Sets and Elementary Mathematics", " Subsets", " The Universal Set", " The Intersection of Sets", " Union of Sets", " Partitioning of Sets", " Subtraction of Sets", " The Algebra of Sets", " Boolean Algebras", "3. The Cardinality of Sets", " Equality Between Sets", " On the Dance Floor", " For Every Flow There Is an Ebb", " Can a Part Be Equal to the Whole?", " Countable Sets", " Algebraic Numbers", " Unequal Sets", " The Countable Set - The Smallest of the Infinite Sets", " Uncountable Sets", " The Census That Never Took Place", " The Uncountability of the Continuum", " The Existence of Transcendental Numbers", " Long and Short Line Segments Have Equally Many Points", " Segment and Square", " Somehow One Problem Does Not Work Out", " Is There a Set of Largest Cardinality?", " The Arithmetic of the Infinite", " Infinite Exponents", " On the Ordering of Numbers", " Completely Ordered Sets", " The Enigmatic Axiom", " Two Apples from One", "4. Remarkable Functions and Curves, or a Stroll through a Mathematical Art Museum", " How the Notion of Function Developed", " The Genie Escapes from the Bottle", " Wet Points", " The Devil's Staircase", " A Prickly Curve", " A Closed Curve of Infinite Length", " A Mathematical Carpet", " Euclid Does Not Rely on Euclid", " Are Rigorous Definitions Needed?", " A Curve Is the Path of a Moving Point", " The Theorem Is Obvious, but the Proof Is Not", " A Curve Passing through All the Points of a Square", " Everything Had Come Unstrung", " How to Make a Statue", " Continua", " Cantor Curves", " Can the Area of a Curve Be Different from Zero?", " Domains without Area", " Some Surprising Examples", " Domains and Boundaries", " The Great Irrigation Project", " A "Nondissertable" Subject", " The Inductive Definition of Dimension", " The Article Is to Be Printed, Not Reviewed!", "Conclusion", "Exercises and Examples".

Hmm, ifølge de gamle grækere kan man ikke dele en cirkel i to ens dele, for centrum måtte jo være i en af dem. Og Stanislav Lem har en variant af Hilberts Hotel. Udvalgsaksiomet medfører at man kan lave ikke-målelige mængder, så simple flytninger i rummet giver Banach-Tarski paradokset med at lave et æble om til to. Russells paradoks med barberen, der barberer alle, der ikke barberer dem selv.

Mange små søde historier og anekdoter. Fx om David Hilbert, der brokker over at en artikel af Urysohn ikke er kommet med i et nummer af Mathematische Annalen og får at vide at den er under review. Men han havde jo sagt: "The Article Is to Be Printed, Not Reviewed!". Så kan noget ikke blive mere godkendt i denne verden!… (mehr)