  As a first step we will restrict our attention to theories which are formulated in a common language and our language of choice will be the language of set theory. The reason for restricting to a common language is that otherwise there will be theories which are merely notational variants of one another. And the reason for focusing on the language of set theory is that it is a universal language in the sense that any system of mathematics is mutually interpretable with a theory formulated in the language of set theory; for example, PA is mutually interpretable with ZFC−Infinity. We are thus setting aside the question of pluralism as it might arise in a more metaphysical setting where one considers questions such as whether natural numbers are “really” sets or whether instead there are two distinct domains of objects—natural numbers and the finite ordinals of set theory. Our focus is on the question of pluralism as it arises in response to the independence phenomenon and the issues at play in this discussion are invariant across different stances on the above metaphysical question. With regard to our question there is no loss of generality in the restriction to the language of set theory since the problem of finding new axioms for PA translates (through the natural interpretation of number theory into set theory) into the problem of finding new axioms for ZFC−Infinity, and conversely. Nevertheless, in what follows we shall, largely out of convenience and familiarity, continue to speak of theories that are formulated in other languages—such as Q and PA—but the reader should always understand us to be strictly speaking of the corresponding theory in the language of set theory. 