Numerical Abstraction via the Frege Quantifier

Notre Dame Journal of Formal Logic 51 (2):161-179 (2010)
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 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
Keywords cardinality quantifiers   abstraction principles   arithmetic
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