Azriel Levy

On Ackermann's Set Theory

Ackermann introduced in [1] a system of axiomatic set theory. The quantifiers of this set theory range over a universe of objects which we call classes. Among the classes we distinguish the sets. Here we shall show that, in some sense, all the theorems of Ackermanns set theory can be proved in Zermelo-Fraenkels set theory. We shall also show that, on the other hand, it is possible to prove in Ackermanns set theory very strong theorems of the Zermelo-Fraenkel set theory.

A Generalization of Godel's Notion of Constructibility

The notion of constructibility introduced by Godel in [2] has been generalized by Hajnal [3], [4] in order to prove the conditional independence of the generalized continuum hypothesis and related axioms. A very similar construction was used independently by Shoenfield [11], [12] 2 and the author [7] to prove the conditional independence of V = L and related axioms. Here we shall prove results further in the latter direction than those in Shoenfield [12].

Alfred Tarski's Work in Set Theory

Alfred Tarski started contributing to set theory at a time when the Zermelo-Fraenkel axiom system was not yet fully formulated and as simple a concept as that of the inaccessible cardinal was not yet fully defined. At the end of Tarskis career the basic concepts of the three major areas and tools of modern axiomatic set theory, namely constructibility, large cardinals and forcing, were already clearly defined and were in the midst of a rapid successful development. The role of Tarski in this development was somewhat similar to the role of Moses showing his people the way to the Promised Land and leading them along the way, while the actual entry of the Promised Land was done mostly by the next generation. The theory of large cardinals was started mostly by Tarski, and developed mostly by his school. The mathematical logicians of Tarskis school contributed much to the development of forcing, after its discovery by Paul Cohen, and to a lesser extent also to the development of the theory of constructibility, discovered by Kurt Godel. As in other areas of logic and mathematics Tarskis contribution to set theory cannot be measured by his own results only; Tarski was a source of energy and inspiration to his pupils and collaborators, of which I was fortunate to be one, always confronting them with new problems and pushing them to gain new ground. Tarskis interest in set theory was probably aroused by the general emphasis on set theory in Poland after the First World War, and by the influence of Wactaw Sierpinski, who was one of Tarskis teachers at the University of Warsaw. The very first paper published by Tarski, [21], was a paper in set theory.