The first edition of this book was published exactly half a century after
the year (1908) in which set theory in its original, naive form created by
Cantor, gravely shaken by the antinomies, underwent a thorough reconstruction
in the hands of Brouwer, Russell, and Zermelo. During the thirteen years
that have passed since the publication of that edition there have been many
developments in most of the topics that were discussed in it. But only in the
axiomatic foundations have there been such extensive, almost revolutionary,
developments as to warrant the almost complete rewriting of Chapter I1 by
one of the authors (A.L.). The senior author, A.A. Fraenkel, had left upon his
death in 1966, extensive notes for the updating of Chapter IV on the intuitionistic
conceptions of mathematics, but it was deemed necessary to invoke
further help, since neither of the other authors felt himself on firm enough
ground in this respect. We would like to thank Dr. D. van Dalen for his
readiness to take upon himself, on short notice, this task, in connection with
which Professor Heyting’s helpful comments should gratefully be acknowledged.