axiom of replacement, it is unable to prove the existence of cardinals greater than ℵω. In PM ordinals are treated as equivalence classes of well-ordered sets, and as with cardinals there is a different collection of ordinals for each type. In ZFC there is only one collection of ordinals, usually defined as von Neumann ordinals. One strange quirk of PM is that they do not have an ordinal corresponding to 1, which causes numerous unnecessary complications in their theorems. The definition of ordinal exponentiation αβ in PM is not equivalent to the usual definition in ZFC and has some rather undesirable properties: for example, it is not continuous in β and is not well ordered (so is not even an ordinal). The constructions of the integers, rationals and real numbers in ZFC have been streamlined considerably over time since the constructions in PM. Differences between editions Apart from corrections of misprints, the main text of PM is unchanged between the first and second editions. The main text in Volumes 1 and 2 was reset, so that it occupies fewer pages in each. In the second edition, Volume 3 was not reset, being photographically reprinted with the same page numbering; corrections were still made. The total number of pages (excluding the endpapers) in the first edition is 1,996; in the second, 2,000. Volume 1 has five new additions: A 54-page introduction by Russell describing the changes they would have made had they had more time and energy. The main change he suggests is the removal of the controversial axiom of reducibility, though he admits that he knows no satisfactory substitute for it. He also seems more favorable to the idea that a function should be determined by its values (as is usual in current mathematical practice). Appendix A, numbered as *8, 15 pages, about the Sheffer stroke. Appendix B, numbered as *89, discussing induction without the axiom of reducibility. Appendix C, 8 pages, discussing propositional functions. An 8-page list of definitions at the end, giving a much-needed index to the 500 or so notations used. In 1962, Cambridge University Press published a shortened paperback edition containing parts of the second edition of Volume 1: the new introduction (and the old), the main text up to *56, and Appendices A and C. Editions The first edition was reprinted in 2009 by Merchant Books, , , . See also Axiomatic set theory Boolean