A Dilemma in the Philosophy of Set Theory

Abstract

We show that the following conjecture about the universe V of all sets is wrong: for all set-theoretical (i.e., first order) schemata $\Sigma$ true in V there is a transitive set $\Sigma$ "reflecting" $\Sigma$ in such a way that the second order statement $\sigma$ corresponding to $\Sigma$ is true in $u$. More generally, we indicate the ontological commitments of any theory that exploits reflection principles in order to yield large cardinals. The disappointing conclusion will be that our only apparently good arguments for the existence of large cardinals have bad presuppositions.