Closing the Hole Argument

The hole argument purportedly shows that spacetime substantivalism implies a pernicious form of indeterminism. Here we attempt to answer the question: what is the mathematical fact that is supposed to underwrite the hole argument? We identify two relevant mathematical claims. The first claim is trivially true, and Weatherall has convincingly argued that it cannot support the hole argument. The second claim would support the hole argument, but we prove that it is false. Therefore, there is no basis for the hole argument.