Tuesday, September 6, 2016

Contingent pure sets?

It is widely held that some sets exist contingently. The standard examples are sets that have contingent entities among their members (or the members of their members or ...), such as the singleton set of me or the set of all actual cows. I wonder if such examples exhaust the contingently existing sets. Could there be contingently existing pure sets, sets whose members all the way down are sets?

Well, they're not going to be sets whose existence can be proved from the axioms of set theory, if these axioms are necessary truths. But one interesting class of potential candidates could be sets defined in non-set-theoretic terms. For instance, suppose that in the actual world a coin is actually tossed an infinite number of times, with the occasions numbered 1,2,3,.... Then, if probability theory is to be applicable to the real world, we need to suppose something like the hypothesis that there is a set of all natural numbers corresponding to occasions when the coin landed heads. But would that set exist if the coin had landed in a radically different sequence or not been tossed at all? I used to assume that of course the answer to a question like that would be affirmative. I still think it's likely to be affirmative. But the matter is far from clear to me now.

1 comment:

Ripheus23 said...

How has no one pursued this question? I mean, except maybe in causal set theory, although that depends on how vague the boundary between a pure set and an impure one is, in the limit... But yeah, the issue of contingently empty sets, for example, immediately arose for me in connection with the question of deontic sets. I mean, imagine some element-and-set X and Y such that OB(X ∈ Y), and X is at worst an ur-element, but still otherwise "purely set-theoretic." If OB implies contingently, as is often supposed, then it would be contingent that X was an element of Y, and in a universe of discourse with no other such things, then if Y was empty of X, Y would be empty period, ergo... But I'm not finding any analysis of this question anywhere else so far.