By Paul Bernays

Since the start of the 20th century, set conception, which started with Euclid's Elements and was once revived within the 19th century, has develop into more and more vital in just about all parts of arithmetic and common sense. partly I of this wonderful monograph, A. A. Fraenkel provides an creation to the unique Zermelo-Fraenkel kind of set-theoretic axiomatics and a historical past of its next development.
In half II Paul Bernays deals an self reliant presentation of a proper approach of axiomatic set thought, masking such issues because the body of common sense and sophistication idea, basic set idea, transfinite recursion, finishing axioms, cardinal mathematics, and strengthening of the axiom system.
The ebook is directed to the reader who has a few acquaintance with difficulties of axiomatics and with the traditional tools of mathematical good judgment. No targeted wisdom of set concept and its axiomatics is presupposed. Readers will locate the severe equipment behind the booklet particularly invaluable. It comprises indexes of authors, symbols and concerns, an inventory of axioms, and an intensive bibliography.

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Exhibiting the main features of these systems would require a lengthy exposition. This is superfluous at this juncture since the main part of the present monograph is just a modified and improved elaboration of ideas which are found in Bernays' papers mentioned above. Therefore the only purpose of the present section is to point out the principles and tendencies of von Neumann, Bernays, Godel in comparison with the system Z exhibited thus far. We shall, however, ignore von Neumann's peculiar terminology.

On the other hand something of the idea of type theory is retained in order that we are not obliged to refrain from forming assemblages which are conceptually natural as for instance that of all sets or all ordinals. Indeed, the concept of the set of all sets - provided we are allowed to apply to such a set the usual formation processes - would give a contradiction, but we can instead operate with the class of all sets.

Pointing to the distinction between propositional functions and attributes. We want to have the classes related to propositional functions, - though not in a way of contextual definition but by a general rule of conversion which we regard as belonging to the logical grammar. Thus we are induced to employ class variables only as free variables, what in fact is sufficient for giving our 1) An other proposal for overcoming the difficulty of the antinomies was made by Heinrich Behmann [1931], which consists in admitting the possibility that the logical functions for some arguments yield a senseless value.

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