Cardinal and Ordinal Numbers

Topics

After five introductory chapters on naive set theory and set-theoretic notation, and a sixth chapter on the axiom of choice, the book has four chapters on cardinal numbers, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets, order types, well-orders, ordinal numbers, ordinal arithmetic, and the Burali-Forti paradox according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern aleph numbers and the continuum hypothesis, statements equivalent to the axiom of choice, and consequences of the axiom of choice.

The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof by Paul Cohen of the independence of the continuum hypothesis, and the construction by Robert M. Solovay of the Solovay model in which all sets of real numbers are Lebesgue measurable.