Gabbay Dov M. - Sets and Extensions in the Twentieth Century

Handbook of the History of Logic, Vol. 6, suppl (C), 2012

ISSN: 1874-5857

doi: 10.1016/B978-0-444-51621-3.50012-8

Front matter

Handbook of the History of Logic, Vol. 6, suppl (C), 2012

ISSN: 1874-5857

doi: 10.1016/B978-0-444-51621-3.50013-X

Copyright

Handbook of the History of Logic, Vol. 6, suppl (C), 2012

ISSN: 1874-5857

doi: 10.1016/B978-0-444-51621-3.50014-1

Preface

Handbook of the History of Logic, Vol. 6, suppl (C), 2012

ISSN: 1874-5857

doi: 10.1016/B978-0-444-51621-3.50015-3

Contributors

Handbook of the History of Logic, Vol. 6, suppl (C), 2012

ISSN: 1874-5857

doi: 10.1016/B978-0-444-51621-3.50001-3

Set Theory from Cantor to Cohen

Akihiro Kanamori

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf.

What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The whole transfinite landscape can be viewed as having been articulated by Cantor in significant part to solve the Continuum Problem. Zermelos axioms can be construed as clarifying the set existence commitments of a single proof, of his Well-Ordering Theorem. Set theory is a particular case of a field of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of set. There were two main junctures, the first being when Zermelo through his axiomatization shifted the notion of set from Cantors range of inherently structured sets to sets solely structured by membership and governed and generated by axioms. The second juncture was when the Replacement and Foundation Axioms were adjoined and a first-order setting was established; thus transfinite recursion was incorporated and results about all sets could established through these means, including results about definability and inner models. With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength. Throughout, the subject has not only been sustained by the axiomatic tradition through Gdel and Cohen but also fueled by Cantors two legacies, the extension of number into the transfinite as transmuted into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory. All this can be regarded as having a historical and mathematical logic internal to set theory, one that is often misrepresented at critical junctures in textbooks (as will be pointed out). This view, from inside set theory and about itself, serves to shift the focus to those tensions and strategies familiar to mathematicians as well as to those moves, often made without much fanfare and sometimes merely linguistic, that have led to the crucial advances.