Abstract: Beginning with Aristotle, and continuing well into the nineteenth century, major philosophers and mathematicians rejected the notion of the actual infinite. They argued that the only sensible notion is that of potential infinity. Closely related to the notion of potential infinity here is that of potential existence. For Aristotle, the points interior to a line segment only exist potentially—they are places where the line can be broken.
It is, we suggest, natural to explicate this notion of potentiality in terms of modal logic. Potentiality, understood modally, is acting here as a kind of guard-rail against paradox—initially the well-known Zeno paradoxes.
The theme of this paper is to see how well this modal conception of potential infinity stands up when a Judeo-Christian deity enters the picture. If God “sees” (or otherwise comprehends) all the points there are in a given line—all of the places where the line could be cut—then these points are on the line even before the cuttings take place. If God “sees” something, it must be there; it must be real or actual in some robust sense.
The nineteenth and early twentieth centuries saw a definitive change in orientation towards the infinite, culminating in the pioneering work of Georg Cantor, who showed us how to make mathematical sense of completed infinite collections or sets. Cantor argued for the exact opposite of the traditional view, stating that the potentially infinite is dubious unless it is somehow backed by an actual infinity.
At least on the surface, however, Cantor was not entirely consistent in his rejection of the potential infinite. Sometimes he ascribed to the so-called “absolutely infinite”, or what he dubbed “inconsistent multitudes” (e.g., the ordinals), features closely analogous to those of the potentially infinite. And, often, the discussion is theological.
The situation for the Cantorian is starkly different from that of the traditional view that rejects all actual infinities. For the Cantorian, the invocation of the deity does not merely threaten to collapse the potentially infinite into the actually infinite; it threatens inconsistency. Even for God, a set of all sets, all cardinal numbers, or all ordinals, is a contradiction.
Let an extending operation be an operation that maps a plurality of objects to a larger (in some sense) such plurality. For example, take some (ordinal) numbers and produce the least upper bound of them. This is Cantor’s “second principle of number generation.”
Extending operations pose a threat if one desires to have collections that cannot be increased. Cantor talks about the multiplicity of all sets, the extended number sequence, and the like (albeit with the qualification that these totalities are “inconsistent”). If one of these totalities could serve as an argument of an extending operation, that would yield yet another object of the relevant kind, and we would have a contradiction: we would have shown that an alleged totality is not a totality after all.
At least initially, potentiality provides an appealing way to deny this availability and thus defuse the threat. For the problematic cases, at least, the extending operation cannot be applied because there is no time or possible world at which the entire plurality of objects that would serve as an argument is available. Instead, the objects that would collectively make up the desired argument are “spread out” with respect to some dimension of increase (time, modality). This is the sense in which potentiality is a “guard-rail” against paradox.
But it is dangerous to bring God into the picture. Surely, from God’s point of view, all of the objects that would collectively make up the disputed argument of an extending operation—say, all points on a line segment, or all sets, or all (ordinal) numbers—must be available. What then prevents God from applying the extending operation, even if we mortals cannot?
We shall find efforts to install a second guard-rail by denying that the extending operation can be applied—some absolute limit on its scope. Not even God can form a single collection of all the relevant objects or produce a least upper bound. Galileo, Leibniz, and their contemporary heirs try versions of this strategy. A better option is to articulate the relevant modality in a way that it becomes plausible that even for God, there is no index that has all of the relevant arguments. This sets up a framework that allows the collapse to be resisted at its very introduction.
The talk will be held on Zoom. Contact Laura.Crosilla @ ifikk.uio.no for a Zoom-invite.