Axiomatizing Mathematical Conceptualism in Third Order Arithmetic

We review the philosophical framework of mathematical conceptualism as an alternative to set-theoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic. This paper is part of a project whose goal is to make a case that mathematics should be disassociated from set theory. The reasons for wanting to do this, which I discuss in greater detail elsewhere ([22]; see also [19] and [23]), involve both the philosophical unsoundness of set theory and its practical irrelevance to mainstream mathematics. Set theory is based on the reification of a collection as a separate object, an elementary philosophical error. Not only is this error obvious, it also has the spectacular consequence of immediately giving rise to the classical set theoretic paradoxes. Of course, these paradoxes are not derivable in the standard axiomatizations of set theory, but that is only because these systems were specifically designed to avoid them. In these systems the paradoxes are blocked by means of ad hoc restrictions on the set concept that have no obvious intuitive justification, which has led to the development of a large literature of attempted rationalizations (e.g., [2, 3, 6, 9, 10, 11, 12, 13, 15]). The heterogeneity of these efforts attests to the difficulty of this task. For example, from a platonistic perspective it seems impossible to give a cogent, principled explanation of why it should be legal to form power sets of infinite sets, given that unrestricted comprehension (forming the set of all x such that P (x)) is not supposed to be valid in general. Antiplatonistic attempts to justify set theory, on the other hand, appear doomed from the start because of the massive gap in consistency strength between straightforwardly antiplatonistically justifiable systems like Peano arithmetic and, say, Zermelo-Frankel set theory. The fact that modern mathematics apparently rests on this kind of basis must be considered a major embarassment for the subject. Probably the real appeal of set theory comes not from any murky philosophical defense, but rather from the role it plays as the standard foundation for mathematics. However, its concordance with normal mathematical practice is actually quite poor. Cantorian set theory postulates a vast universe of sets containing remote cardinals which bear no relation to the relatively concrete world of ordinary mathematics, where most objects of central interest are essentially countable (i.e., separable for some natural topology). Similarly, set theory as a mathematical discipline is quite isolated from the rest of mathematics, and it could hardly be otherwise