## Friday, February 20, 2015

### A cardinality objection to unrestricted modal profiles

The modal profile of an object tells us which worlds the object exists in and what it consists of in those worlds.

The unrestricted modal profiles (UMP) thesis says that for any map f that assigns to some worlds w a concrete object f(w) in w and that assigns nothing to other worlds, there is a possible concrete object Of such that Of exists in all and only the worlds w to which f assigns an object and has the property that in w, Of is wholly composed of f(w) (or of parts of f(w) that compose f(w)).

One can think of UMP as the next step after unrestricted composition (UC) which holds that for any concrete objects there is an object composed of them. UC is not enough to guarantee the existence of ordinary objects like tables and chairs, since there is no guarantee that the modal profile of a UC-guaranteed object composed of the particles in a table will match the modal profile of a table. But UC+UMP, plus a thesis about the physical world being made of temporal parts of particles, will give us what we need here.

However, UMP is false.

1. There is a set of all actual concrete objects.
2. If UMP is true, then for any cardinality K, there are at least K actual concrete objects.
3. So, if UMP is true, there is no set of all actual concrete objects. (By 2)
4. So, UMP is not true. (By 1 and 3)
Now, claim (1) is very plausible. Claim (3) follows from (2) since for any set there is a greater cardinality by Cantor's Theorem and so it's impossible to have a set whose cardinality is at least as large as every cardinality.

That leaves (2). Assume UMP. Suppose K is any infinite cardinality (we don't need to worry about the finite case). For any K, there is a possible world w with at least K concrete objects (say, K photons). Let x be any concrete object in the actual world @. Then there will be at least K maps f with the property that f(@)=x and f(w) is a concrete object in w and f assigns nothing to any other world. To each such map f there corresponds at least one distinct object Of in the actual world. (Distinct as difference in modal profiles implies non-identity of objects by Leibniz's Law.) So there are at least K concrete objects in the actual world.