Let me start with (a). The mathematical fictionalist has relied on the compactness theorem to formulate the notion of conservativeness in an acceptable way, that is, without reference to mathematical entities. As noted above, conservativeness is defined in terms of consistency. But this notion is usually formulated either in semantic terms (as the existence of an appropriate model), or in proof-theoretic terms (in terms of suitable proofs). However, as Field acknowledges, these two formulations of consistency are platonist, since they depend on abstract objects (models and proofs), and therefore are not nominalistically acceptable.