Williamson's many necessary existents

Abstract

This note is to show that a well-known point about David Lewis’s (1986) modal realism applies to Timothy Williamson’s (1998; 2002) theory of necessary existents as well.1 Each theory, together with certain “recombination” principles, generates individuals too numerous to form a set. The simplest version of the argument comes from Daniel Nolan (1996).2 Assume the following recombination principle: for each cardinal number, ν, it’s possible that there exist ν nonsets. Then given Lewis’s modal realism it follows that there can be no set of all (that is, Absolutely All) the nonsets. For suppose for reductio that there were such a set, A; let ν be A’s cardinality; and let μ be any cardinal number larger than ν. By the recombination principle, it’s possible that there exist μ nonsets; by modal realism, there exists a possible world containing, as parts, μ nonsets; each of these nonsets is a member of A; so A’s cardinality cannot have been ν. On some conceptions of what sets are, Lewis could simply accept this conclusion. But given the iterative conception of set,3 it seems that there must exist a set of all nonsets.4 According to the iterative conception, sets are “built up” in a series of “stages”. At the rst stage a set is “formed” whose members are all and only the nonsets. At subsequent stages, sets are formed whose members are sets from earlier stages. The sets, on this conception, are all and only those that are formed at some stage or other. Since a set of all the nonsets is formed at the very rst stage, such a set must exist. My main concern here is not to defend this argument, only to show how an analogous argument against Williamson may be constructed. Still, it’s worth noting that the recombination principle on which the argument is premised has a solid intuitive basis. Under the broad sense of ‘possible’ at issue, there should be no arbitrary limits to what’s possible; but any limit to how many nonsets are possible would be arbitrary. It would be strange to say that there could