Bulletin of Symbolic Logic

On arbitrary sets and ZFC

José Ferreirós

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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.

Article information

Source
Bull. Symbolic Logic Volume 17, Issue 3 (2011), 361-393.

Dates
First available: 6 July 2011

Permanent link to this document
http://projecteuclid.org/euclid.bsl/1309952318

Digital Object Identifier
doi:10.2178/bsl/1309952318

Zentralblatt MATH identifier
05956503

Mathematical Reviews number (MathSciNet)
MR2856078

Citation

Ferreirós, José. On arbitrary sets and ZFC . Bulletin of Symbolic Logic 17 (2011), no. 3, 361--393. doi:10.2178/bsl/1309952318. http://projecteuclid.org/euclid.bsl/1309952318.


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