First let me head off the suspicion that I’ll be discussing some sort of weird idealist/massively subjectivist/etc metaphysics. I’m concerned with ‘world’ and ‘universe’ as they’re used by analytic metaphysicians, logicians, and philosophers of mathematics. In particular I’m concerned about the cardinalities of the domains of discourse assumed by some philosophers when dealing with, in particular, arguments essentially involving quantification and similar logico-mathematical resources.

In this sense, a world is roughly what we might call a universe along with everything in it in everyday discourse. The universe is the mathematical universe – every mathematical thing one happens to think exists. The question in the title of this post is then a way of raising a gripe about an assumption that often made by philosophers: that the world, and thus the “domain of quantification” is set-sized.

We often speak as if we can quantify over absolutely everything, or at least absolutely every-actual-thing, but then continue to reason as if all of those (actual) things form a set. In many cases this looks perfectly harmless. If we’re talking about medium-sized dry goods, for example, we can think of our quantifiers as being implicitly restricted to e.g. physical objects (our second-order quantifiers to sets of those, etc). As on even the most liberal views of what counts as a physical object, there aren’t more than continuum-many (the cardinality of the real numbers) of them, we shouldn’t run into an immediate problems.

But what if I make the claim (as I often do), “there are abstract objects”. This might naturally be read as “among all of the things there are, one or more is abstract”. If we are assuming that the world is set-sized we’ve ruled out there being enough stuff to correspond to the abstracta I wanted in the first place–mathematical objects. Worse, we can’t even use standard set-theory to reason about the whole world, because, by Cantor’s theorem (the set of all subsets of a set is strictly larger than the original set), we would have to admit that there’s more stuff than there is. That seems problematic to say the least.

Now, maybe you’re sitting there with your nominalist hat on thinking *ok, but there are no abstracta, mathematical objects, etc.* Surely you ought not rule such things out on the grounds that it makes your model theory easier.

Here’s a case that bothers me even more. Let’s stop thinking only about actual things, and start thinking about possibilia as well. In standard Kripkean possible world semantics for modal operators (necessarily and possibly) we start with a “universal”domain of objects, some or all of which are present at each possible world, with the (extensional) predicates being assigned subsets of the powerset of the domain, and an accessibility function that tells which which worlds can “see” which others (functions are also often seen as certain kinds of sets). The problem here is that, if the actual world contains the entire set-theoretic universe (as I think it does), it can’t be a possible world – there are more than set-many sets. This would mean that our semantics of modals can’t tell us anything about whats actually possible, i.e. possible *here*.

Again, whatever you think the status of mathematical objects is, questions about mathematical ontology shouldn’t be forced on us by our semantics of modals.

I could give some more examples, but the idea should be clear. Luckily there is a large and growing literature on absolute generality, indefinite extensibility, general quantification, alternate semantics for modals, etc, which is to say we need not assume that the world is set-sized. There are well worked out alternatives.

The point of all of this is just to encourage us all to have these sorts of issues in mind when we’re working on things where quantifier “ranges”, and cardinality considerations are important.

NB: The scare quotes are mostly there to head off objections from the cognoscenti.

*Logic, Metaphysics, Philosophy, Philosophy of Math*

Posted on June 11, 2016byAaron Thomas-Bolduc