  Predicativism is of particular interest for our present purposes since it grants the frameworks of the natural numbers but rejects most of set theory and, in fact, most of second-order arithmetic. The predicativist is a non-pluralist about first-order arithmetic and thinks that there is an objective fact of the matter concerning every statement formulated in the language of first-order arithmetic. And the predicativist accepts as much of second-order arithmetic and set theory that can be simulated in a theory such as ATR0 (Arithemetical Transfinite Recursion). But the predicativist remains a pluralist about what lies above. The predicativist will often grant that there is utility in employing axiomatic systems of set theory but will maintain that the issue of truth does not arise at that level since there is no univocal underlying conception of set. For example, the predicativist will think that there is an objective choice between PA and PA + ¬Con(PA) but not think that there is an objective choice between PA2 + PU and PA2 + ¬PU (where here PA2 is (schematic) second-order arithmetic) or between ZFC and ZFC + “There are no strongly inaccessible cardinals”. Insofar as the predicativist thinks that the independence results in set theory concerning second-order arithmetic are a reason for endorsing pluralism about both ZFC and PA2 the non-pluralist case we shall present below is directly engaged with the predicativist. But we should stress that for this discussion we shall be presupposing ZFC and so the most direct brand of non-pluralism/pluralism that we shall be engaged with is one that takes a non-pluralist stance toward ZFC but a pluralist stance toward statements like PU which are formally unresolvable on the basis of ZFC. This is a fairly common view (see, for example, Shelah (2003)). 