Subsystems of Quine's ``New Foundations'' with Predicativity Restrictions

Abstract

This paper presents an exposition of subsystems $\mathit{NFP}$ and $\mathit{NFI}$ of Quine's $\mathit{NF}$, originally defined and shown to be consistent by Crabbé, along with related systems $\mathit{TTP}$ and $\mathit{TTI}$ of type theory. A proof that $\mathit{TTP}$ (and so $\mathit{NFP}$) interpret the ramified theory of types is presented (this is a simplified exposition of a result of Crabbé). The new result that the consistency strength of $\mathit{NFI}$ is the same as that of $\mathit{PA}_2$ is demonstrated. It will also be shown that $\mathit{NFI}$ cannot be finitely axiomatized (as can $\mathit{NF}$ and $\mathit{NFP}$).