Alternatives to Set Theory

Classifying objects --- thus collecting them into classes --- is an
operation both fundamental and common in everyday life as well as in
scientific endeavours. Logic, philosophy, mathematics and computer
science analyse this activity by means of such notions as `class',
`set', and `type'. Furthermore, according to the so-called ``logical
(or naive) conception of classes", these notions are related to the
notion `concept' in that classes are considered `extensions' of
concepts. This relationship, however, is a delicate one --- as we
have learned from the paradoxes of class theory. The contributions to
this volume shed new light both on concepts and classes and their
delicate relationship. New and original approaches to these issues
(e.g., map theory, epsilon-sets, intensional type theory, AFA set
theory) are presented and discussed in the broader context of their
application to problems from informatics and linguistics. The focus is
on such matters as unrestrictive comprehension, extensionality, the
relationship between the notions of concept, class and extension, and
on unfounded (circular and infinitely descending) patterns of
elementhood. These and related problems have always played a role in
the development of set theory and have always delivered motivation for
the development of unorthodox views and theories. This is also
demonstrated in this volume by contributions dealing with the history
of Zermelo's axiomatization of set theory and Ackermann's class theory
with unrestrictive comprehension, respectively.
The contributions of this volume are based on the presentation at a
Workshop on ``Alternative Set Theories --- Alternatives to Set Theory"
held at the University of Southern Denmark in 2006.