I should now consider issue (b): Field's nominalistic proof of the conservativeness of set theory. Let us grant that the concept of conservativeness has been formulated in some nominalistically acceptable way. If Field's proof were correct, he would have proved that mathematics itself is conservative—as long as one assumes the usual reductions of mathematics to set theory. How does Field prove the conservativeness of set theory? It is by an ingenious argument, which adapts one of the Field's platonistic conservativeness proofs (Field 1980). For our present purposes, we need not examine the details of this argument, but simply note that at a crucial point the completeness of first-order logic is used to establish its conclusion (Field 1992, p. 118).