Leibniz in Cantor’s Paradise: A Dialogue on the Actual Infinite

Abstract

In this paper I present a fictional dialogue between Gottfried Leibniz and Georg Cantor on the actual infinite. The dialogue is set in the afterlife, and I use the authors’ own words to the extent I can. Leibniz, enlisting a distinction due to the Scholastics, denied the actual infinite in the sense of a collection or set of terms (the categorematic infinite) in favour of a syncategorematic understanding: an actually infinite multiplicity of terms, syncategorematically understood, is one such that however many one supposes there to be, there are more; but there is no infinite number of them. When referring to “all” the terms in an infinite multiplicity, the “all” must be understood distributively, not collectively; Leibniz consistently rejects infinite collections or totalities as being incompatible with the Part-Whole axiom. One consequence of this is to show that Cantor’s diagonal argument does not constitute a conclusive proof that there are more reals than natural numbers, since it depends on the premise that the real numbers form a totality, a collection such that none are left out; and this is established by the Power Set Axiom only on the hypothesis that the natural numbers form such a totality. Similarly, 1–1 correspondence between the elements of two multiplicities does not establish that they constitute equal sets without the assumption that those infinite multiplicities are indeed consistent totalities. Consistency, on the Leibnizian model, would have to be shown by the provision of “real definitions”, thus committing him to a kind of constructivism that would rule out Cantorian transfinite recursion. In contrast, it is shown that Leibniz’s syncategorematic understanding of the actual infinite is not only consistent with this constructivism, but also with his own conception of the actually infinite division of matter, whereas Cantor’s transfinite is not. Lastly, it is shown that Leibniz’s claim that the universe, or any other collection of all unities, cannot itself be a unity, can be proved in an entirely analogous way to Cantor’s proof that there is no ordinal number of all ordinal numbers. In sum, I use this dialogue to argue that the Leibnizian actual infinite constitutes a perfectly clear and consistent third alternative in the foundations of mathematics to the usual dichotomy between the potential infinite (Aristotelianism, intuitionism) and the transfinite (Cantor, set theory), and one that avoids the paradoxes of the infinite.