Explicit Abstract Objects in predicative settings

Abstractionist programs in the philosophy of mathematics have focused their investigations on abstraction principles, which are taken to be implicit definitions of the objects in the range of their operators. Such principles are consistent in predicative second-order logic (\SOL), but are (individually) mathematically weak. This paper, inspired by the work of Boolos \cite{Boolos1986} and Zalta \cite{Zalta1983}, examines axioms of \textit{explicit} definition of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula $\phi(F)$, with $F$ a concept variable. Such a system is inconsistent in full \SOL, but as Zalta shows, can be made consistent with modifications to the logic motivated by a primordial separation of the universe into abstracts and non-abstracts. In this article, we will show that no such measures are needed if the underlying logic is a restrictive version of predicative \SOL. We also show that the axiomatic system we will investigate, i.e. \RPELO, has a natural extension which delivers a peculiar interpretation of \PA$^2$.