Numerical Abstraction via the Frege Quantifier

Abstract

This paper presents a formalization of rst-order arith- metic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interac- tion of a non-standard (but still rst-order) cardinality quantier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a non-reductionist conception of logicism, a deationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones. | binding one (or more) variables and taking two formulas ' and as arguments, assert- ing that there are no more ''s than 's. Using the Frege quantier it is cleraly possible to dene the equinumerosity of ' and , by say- ing that there and no more ''s than 's and vice-versa. The second device is an abstraction operator assigning objects to predicates (or, as Frege would say, \concepts), with the intended interpretation that the object assigned to ' is | or perhaps represents | the number of objects satisfying the formula '. The interaction of these two devices is most notably captured in the so-called \Hume's Principle (HP), the statement that the number of ''s equals the number of 's if and only if the ''s are equinumerous to the 's.