The Concept of a Set

Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three different kinds of logic (classical, intuitionistic, and minimal). A foundational system incorporating this analysis and based on the principles of mathematical conceptualism [8] accords better with actual mathematical practice than Zermelo-Fraenkel set theory does. 1. Metaphysical interpretations 1.1. Sets in ordinary language. Elementary introductions to set theory tend to give the impression that the concept of a set is trivial, something with which we are already thoroughly familiar from everyday life. We may be told that such things as a flock of birds, a deck of cards, or a pair of apples are examples of sets. This immediately seems strange because sets in the mathematical sense are supposed to be abstract objects not existing in space and time, whereas it is hard to believe that a simple assertion about, say, a flock of birds carries any significant metaphysical content. Slater ([7], Section II) analyzed the way collective expressions are actually used in ordinary language and showed that they indeed express no metaphysical content. A deck of cards exists in space and time; it is a physical object composed of cards in the same way that a house is made out of bricks, something philosophers call a “mereological sum”. It is not a set of cards in the mathematical sense, with cards as elements. The same is true of such things as a bunch of grapes, a herd of cows, etc. (You don’t eat the elements of a bunch of grapes, you eat the bunch of grapes. And “the fact that such collections are mereological sums is also shown by the fact that shoals, herds, packs, tribes, and the like, are located and can move around in physical space, just like their members” ([7], p. 61).) In contrast, the word “pair” in the phrase “pair of apples” is a numerical measure word; to say that you ate a pair of apples merely tells how many apples you ate. It is analogous to the word “half” in the phrase “half a loaf of bread”. And “in ‘There is a half of a loaf’ there is obviously no reference to something other than bread; there is not, in addition, reference to one of a range of mysterious, further objective entities, ‘halves’, ‘quarters’, ‘parts’, etc. There is merely a specification of how much of a loaf there is” ([7], p. 62). The idea that the mathematical concept of a set is obvious and in no need of any special explanation is not correct. This is sometimes noted in the philosophical literature on mathematical foundations, but it seems the conclusion that is usually