The author [A. A. Fraenkel] is well known for his research in set

theory as well as his published textbooks in this subject. He has

previously written the book "Einleitung in die Mengenlehre" which

appeared in three editions. The last edition which was published in

1928 was reprinted in New York in 1946. Whereas "Einleitung in die

Mengenlehre" contained an exposition of classical set theory as well

as a survey of modern theoretical research in the foundations of

mathematics, Fraenkel has now decided to write the present book as an

account of the classical theory only. The modern aspects of foundation

theory will be discussed in another book under the title "Foundations

of Set Theory" which is due to appear about 1955. Presumably, the

reason for this division of the contents of "Einleitung in die

Mengenlehre" into two different books is that the subject matter has

grown too large. The reviewer, however, is not enthusiastic about this

division since such a textbook as the present one will be read

primarily by students and they might form the impression that

classical set theory is securely founded just as other parts of

mathematics, e.g. arithmetic. Such an impression would, however, be

misleading. If it were not so, we could omit the entire modern

foundational research without real loss to mathematics. To the

reviewer it seems unfortunate that classical set theory is developed

in a separate book so that all scruples - or almost all of them - are

reserved for the second volume. This might have the effect that most

readers of this present volume will probably not become acquainted

with the criticisms at all. It is true that some hints to such

scruples are given, but most students might not think that they are

important. On the other hand, it must be conceded that the lack of

knowledge of the results of foundational research will not mean much

to mathematicians who are not especially interested in the logical

development of mathematics.

[Th. Skolem: "Review of: A. A. Fraenkel : Abstract Set Theory.

Amsterdam & Groningen, North-Holland Publishing Company, 1953. XII +

479 pp. " Mathematica Skandinavica 1 (1953) 313.]

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Gruß, WM

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