I was thinking about Patrick Grim’s Cantorian argument just now, and I was trying to think of a really easy way to explain to somebody who isn’t at all familiar with set theory why the argument does establish that there is no such thing as the set of all truths (since such a set is not only infinite but indefinite, because we can take the power set of any set, ergo etc.). Whether cogently related or not, I came to the following thought:
Suppose that in some world W there is only one true proposition X. All other propositions which may have been true of that world are not true. At W, then, only X is true. However, this is obviously logically false. If X is true, then there is at least one other truth, namely proposition P: “X is true”. Moreover, if P is true (which follows from X‘s being true) then there is another truth, namely P1: “P is true”. Eventually one will be able to multiply the set of all truths indefinitely (Pn: “Pn-1 is true”). This seems to entail that there is no logically possible world at which only one proposition is true. In fact, it entails that there is no logically possible world at which any set (finite or infinite) of propositions are true. I’m not sure if this extrapolation from X to Pn: “Pn-1 is true” is perfectly analogous to what is going on when one takes the power set of some set which was supposed to be the set of all truths, but it seems (at least to me) like something very similar is going on.