A Logical Framework for Set Theories

Abstract: Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for Mathematical Knowledge Management. However, in order to be used for this task it is necessary to overcome serious gaps that exist between the "official" formulations of set theory (as given e.g. by formal set theory ZF) and actual mathematical practice. In this work we present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity.