Categorical New Foundations

Abstract

New Foundations ($\mathrm{NF}$) is a set theory obtained from naive set theory by putting a stratification constraint on the comprehension schema; for example, it proves that there is a universal set $V$. $\mathrm{NFU}$ ($\mathrm{NF}$ with atoms) is known to be consistent through its close connection with models of conventional set theory that admit automorphisms. A first-order theory, $\mathrm{ML}_\mathrm{CAT}$, in the language of categories is introduced and proved to be equiconsistent to $\mathrm{NF}$ (analogous results are obtained for intuitionistic and classical $\mathrm{NF}$ with and without atoms). $\mathrm{ML}_\mathrm{CAT}$ is intended to capture the categorical content of the predicative class theory of $\mathrm{NF}$. $\mathrm{NF}$ is interpreted in $\mathrm{ML}_\mathrm{CAT}$ through the categorical semantics. Thus, the result enables application of category theoretic techniques to meta-mathematical problems about $\mathrm{NF}$ -style set theory. For example, an immediate corollary is that $\mathrm{NF}$ is equiconsistent to $\mathrm{NFU} + |V| = |\mathcal{P}(V)|$. This is already proved by Crabb\'e, but becomes more transparent in light of the results of this paper. Just like a category of classes has a distinguished subcategory of small morphisms, a category modelling $\mathrm{ML}_\mathrm{CAT}$ has a distinguished subcategory of type-level morphisms. This corresponds to the distinction between sets and proper classes in $\mathrm{NF}$. With this in place, the axiom of power objects familiar from topos theory can be appropriately formulated for $\mathrm{NF}$. It turns out that the subcategory of type-level morphisms contains a topos as a natural subcategory.