Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T09:13:06.541Z Has data issue: false hasContentIssue false
  • English
  • Français

AN AXIOMATIC THEORY OF WELL-ORDERINGS

Published online by Cambridge University Press:  04 March 2011

OLIVER DEISER
OLIVER DEISER*
Affiliation:
Technische Universität München
*
*TECHNISCHE UNIVERSITÄT MÜNCHEN, SCHOOL OF EDUCATION, SCHELLINGSTR. 33, 80799 MÜNCHEN, GERMANY. E-mail: deiser@tum.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gödel’s axiom of constructibility. In list theory there are strong arguments favoring Gödel’s axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 4 , Issue 2 , June 2011 , pp. 186 - 204
Copyright
Copyright © Association for Symbolic Logic 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Blass, A., & Gurevich, Y. (2004). Why Sets? Bulletin of the European Association of Theoretical Computer Science, 84, 139–156.Google Scholar
Cantor, G. (1890). Zur Lehre vom Transfiniten. Halle, Germany: Pfeffer.Google Scholar
Cantor, G. (1897). Beiträge zur Begründung der transfiniten Mengenlehre 1-2. Mathematische Annalen, 49, 207–246.Google Scholar
Cantor, G. (1991). Briefe. In Meschkowski, H., and Nilson, W., editors. Berlin: Springer.Google Scholar
Cohen, P. (1966). Set Theory and the Continuum Hypothesis. Reading, MA: W.A. Benjamin.Google Scholar
Dales, H. G., & Olivieri, G., editors. (1998). Truth in Mathematics. New York, NY: Clarendon Press, Oxford University Press.Google Scholar
Deiser, O. (2006). Orte, Listen, Aggregate. FU Berlin: Habilitationsschrift.Google Scholar
Deiser, O. (2008). Reelle Zahlen. Berlin: Springer.Google Scholar
Deiser, O. (2009). Einführung in die Mengenlehre. Berlin: Springer.Google Scholar
Deiser, O. (2010). On the development of the notion of a cardinal number. History and Philosophy of Logic, 31/2, 123–144.Google Scholar
Ebbinghaus, H.-D. (2007). Ernst Zermelo. An Approach to his Life and Work. Berlin: Springer.Google Scholar
Ferreirós, J. (1999). Labyrinths of Thought. A History of Set Theory and its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser.Google Scholar
Frege, G. (1967). Kleine Schriften. Hildesheim, Germany: Olms.Google Scholar
Frege, G. (1983). Nachgelassene Schriften und wissenschaftlicher Briefwechsel. Nachgelassene Schriften (second edition). Vol. 1. Wissenschaftlicher Briefwechsel. Vol. 2. Hamburg, Germany: Meiner.Google Scholar
Frege, G. (1986). Grundlagen der Arithmetik. Hamburg, Germany: Meiner.Google Scholar
Gericke, H. (1970). Geschichte des Zahlbegriffs. Mannheim, Germany: Bibliographisches Institut.Google Scholar
Gödel, K. (1938). The consistency of the axiom of choice and the generalized continuum hypothesis. Proceedings of the National Academy of Sciences USA, 24, 556–557.Google Scholar
Gödel, K. (1939). Consistency-proof for the generalized continuum-hypothesis. Proceedings of the National Academy of Sciences USA, 25, 220–224.CrossRefGoogle ScholarPubMed
Hausdorff, F. (1914). Grundzüge der Mengenlehre. Leipzig, Germany: Veit & Comp.Google Scholar
Heath, T. (2003). A manual of Greek Mathematics. Dover, NY: Oxford University Press.Google Scholar
Jensen, R. (1995). Inner models and large cardinals. The Bulletin of Symbolic Logic, 1/4, 393–407.Google Scholar
Kanamori, A. (1996). The mathematical development of set theory from Cantor to Cohen. The Bulletin of Symbolic Logic, 2/1, 1–71.Google Scholar
Kanamori, A. (2003). The empty set, the singleton, and the ordered pair. The Bulletin of Symbolic Logic, 9/3, 273–298.Google Scholar
von Neumann, J. (1923). Zur Einfühhrung der transfiniten Zahlen. Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum 1922/1923, Szeged, 199–208.Google Scholar
Woodin, W. H. (2001). The continuum hypothesis, part I and II. Notices of the American Mathematical Society, 48, 567–576(part I), 681–690 (part 2).Google Scholar
Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59, 514–516.CrossRefGoogle Scholar
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre. I. Mathematische Annalen, 65, 261–281.CrossRefGoogle Scholar