A Paradox of ZF-Class Nominalism

In a recent article in this journal, Calemi challenges the Küng-Armstrong trilemma, a well-known objection to traditional class nominalism, by proposing a fusion of class nominalism with Zermelo-Fraenkel set theory (ZF). In this note, we argue that ZF-class nominalism faces significant challenges in the form of incompleteness and potential paradoxes stemming from Gödel’s incompleteness theorem. We will explore these issues in detail, highlighting the key implications for the viability of ZF-class nominalism as a philosophical position.