Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?

Abstract

Abstract. Cantor's theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same 'size' in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the 'size' of A should be less than the 'size' of B (part-whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part-whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part-whole principle, envisaged by Bolzano (Kitcher). §1. Introduction. Two central issues seem to have determined the reflection on mathematical infinity in Western thought. The first concerns its existence. The second whether it can be measured. In this paper, I will only deal with the second aspect of the issue although, of course, the two issues cannot always be separated. The structure of the paper is as follows. First, I will retrace some of the major historical positions that were taken with respect to the paradoxical properties displayed by infinite sets of natural numbers with emphasis on whether there could be an arithmetic of infinite sets. In the second part, I will describe recent mathematical developments that offer a way to measure the size of infinite sets of natural numbers while preserving the part-whole principle. In the conclusion, I will offer some philosophical reflections as to how these recent mathematical developments impact various historical and philosophical claims found in the literature, including Gödel's claim, which I contest, as to the inevitability of Cantor's definition of infinite number. This is the notion of inevitability referred to in the title of this paper, namely the claim that if one wants to generalize the notion of number from the finite to the infinite there is only one possible way to go and that is the Cantorian notion of cardinal number.

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