On arbitrary sets and ZFC

Bulletin of Symbolic Logic 17 (3):361-393 (2011)
  Copy   BIBTEX

Abstract

Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo—Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,221

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Sets and Point-Sets: Five Grades of Set-Theoretic Involvement in Geometry.John P. Burgess - 1988 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:456 - 463.
Finite mathematics.Shaughan Lavine - 1995 - Synthese 103 (3):389 - 420.
The ordertype of β-r.E. Sets.Klaus Sutner - 1990 - Journal of Symbolic Logic 55 (2):573-576.
Anti-admissible sets.Jacob Lurie - 1999 - Journal of Symbolic Logic 64 (2):407-435.
Almost weakly 2-generic sets.Stephen A. Fenner - 1994 - Journal of Symbolic Logic 59 (3):868-887.
Presburger sets and p-minimal fields.Raf Cluckers - 2003 - Journal of Symbolic Logic 68 (1):153-162.
Strong measure zero sets without Cohen reals.Martin Goldstern, Haim Judah & Saharon Shelah - 1993 - Journal of Symbolic Logic 58 (4):1323-1341.
Ramsey sets, the Ramsey ideal, and other classes over R.Paul Corazza - 1992 - Journal of Symbolic Logic 57 (4):1441 - 1468.

Analytics

Added to PP
2011-07-07

Downloads
103 (#155,554)

6 months
10 (#134,868)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Jose Ferreiros
Universidad de Sevilla

Citations of this work

Predicativity and Feferman.Laura Crosilla - 2017 - In Feferman on Foundations. Springer Verlag. pp. 423-447.
Frege, Dedekind, and the Origins of Logicism.Erich H. Reck - 2013 - History and Philosophy of Logic 34 (3):242-265.
Dedekind’s Map-theoretic Period.José Ferreirós - 2017 - Philosophia Mathematica 25 (3):318–340.

View all 7 citations / Add more citations

References found in this work

Philosophy of Mathematics and Natural Science.Hermann Weyl - 1949 - Princeton, N.J.: Princeton University Press. Edited by Olaf Helmer-Hirschberg & Frank Wilczek.
Naturalism in mathematics.Penelope Maddy - 1997 - New York: Oxford University Press.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press USA.
From Frege to Gödel.Jean Van Heijenoort (ed.) - 1967 - Cambridge,: Harvard University Press.
Introduction to Mathematical Philosophy.Bertrand Russell - 1919 - Revue Philosophique de la France Et de l'Etranger 89:465-466.

View all 51 references / Add more references