Notre Dame Journal of Formal Logic

Numerical Abstraction via the Frege Quantifier

G. Aldo Antonelli


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.

Article information

Notre Dame J. Formal Logic Volume 51, Number 2 (2010), 161-179.

First available in Project Euclid: 11 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C99: None of the above, but in this section 03C80: Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48]

cardinality quantifiers abstraction principles arithmetic


Antonelli, G. Aldo. Numerical Abstraction via the Frege Quantifier. Notre Dame J. Formal Logic 51 (2010), no. 2, 161--179. doi:10.1215/00294527-2010-010.

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