However, how are such judgments of impossibility established? Under what conditions do we know that they hold? In simple cases, involving straightforward statements, to establish such judgments may be unproblematic. The problem emerges when more substantive statements are invoked. In these cases, we seem to need a significant amount of mathematical information in order to be able to determine whether the impossibilities in question really hold or not. Consider, for instance, the difficulty of establishing the independence of the axiom of choice and the continuum hypotheses from the axioms of Zermelo-Fraenkel set theory. Significantly complex mathematical models need to be constructed in this case, which rely on the development of special mathematical techniques to build them. What is required from the mathematical fictionalist at this stage is the nominalization of set theory itself—something that, as we saw, Field still owes us.