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.

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