On arbitrary sets and ZFC
|Author||Ferreirós Domínguez, José Manuel|
|Department||Universidad de Sevilla. Departamento de Filosofía y Lógica y Filosofía de la Ciencia|
|Published in||The Bulletin of Symbolic Logic, 17(3), 361-393|
|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 ...
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 maximalist. 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 too ﬀeran 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.