ABSTRACT
Neo-logicism faces two challenges. First, the neo—logicist must solve the Bad Company Objection-that is, she must provide a principled account that separates those abstraction principles that are “good” (e.g., Hume’s Principle) from those abstraction principles that are “bad” (e.g., Basic Law V). Second, if neo-logicism is to provide a foundation for all of (classical) mathematics, then the neo-logicist needs to provide a viable reconstruction of set theory. In this chapter, we show that any neo-logicist account of set theory will entail the axiom of complement. A simple set-theoretic paradox is used to show that no consistent theory of sets can contain both the axiom of separation and the axiom of complement. The argument for complement (and hence against separation) then proceeds by carefully considering what a successful account of neo-logicist set theory must look like (the global approach to sets) and arguing that any such account of set theory will, when combined with any of the extant suggestions for solving the Bad Company Problem (or any of a wide variety of variations on such suggestions) entails the axiom of complement.
