This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard (but still first-order) cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.
"Numerical Abstraction via the Frege Quantifier." Notre Dame J. Formal Logic 51 (2) 161 - 179, 2010. https://doi.org/10.1215/00294527-2010-010