Frege's logicist program requires that arithmetic be reduced to
logic. Such a program has recently been revamped by the
"neologicist" approach of Hale and Wright. Less attention has
been given to Frege's extensionalist program, according to which
arithmetic is to be reconstructed in terms of a theory of
extensions of concepts. This paper deals just with such a theory.
We present a system of second-order logic augmented with a
predicate representing the fact that an object x is the
extension of a concept C, together with extra-logical axioms
governing such a predicate, and show that arithmetic can be
obtained in such a framework. As a philosophical payoff, we
investigate the status of the so-called Hume's Principle and its
connections to the root of the contradiction in Frege's system.
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