Wednesday, February 10, 2016

Cardinality and worlds

For every initial ordinal k, there is a possible world with exactly k photons. But there is no set of all initial ordinals (proof: suppose there is such a set; the union of the members of any set of ordinals is an ordinal; so the union of the initial ordinals is an ordinal; it must have the same cardinality as some initial ordinal in the set; but for any ordinal in the set, there is a larger one in the set). So there is no set of all possible worlds.

This argument doesn't use the Axiom of Choice and hence it improves on the argument I gave here.


Heath White said...

What is the upshot of this result? (Serious question.) Maybe it is problematic if you said, "For every member of the set of possible worlds, Pv~P." But you can just rephrase that to "For all possible worlds..." or "For any possible world..." and make the same claim. So what is at stake?

Alexander R Pruss said...

In the paper I link to, it's meant to be a substantive problem for Lewis's modal realism, because plausibly for any type T of material object, there is a set of all the Ts. According to Lewis, worlds are material objects. So there should be a set of all the worlds.

It's also a technical (though maybe not substantive) problem for anyone wanting to use sets of possible worlds in philosophical accounts. For instance, if there is no set of all worlds, then propositions aren't sets of the worlds at which they are true.

Moreover, even if propositions aren't sets of worlds, since there are at least as many propositions as worlds (for every world, there is the contingent proposition that that world is actual), it follows that there is no set of all propositions. This creates technical (though maybe not substantive) problems, too.