Hume’s Principle and the Number of all Objects

Abstract

obtained by adjoining Hume's principle to axiomatic second-order logic, and the name Frege's theorem for the result that 'zero', 'immediately precedes' and 'natural number' can be defined in the language of Frege arithmetic in such a way that the second-order Peano postulates may be proved in that system from those definitions. Of course, the historical Frege did not derive his versions of the Peano postulates in the consistent system that Boolos calls Frege arithmetic. Rather, his derivations were cast in the inconsistent system that is obtained by adjoining to axiomatic second-order logic the notorious Basic Law V. But because the courses-of-values regulated by Basic Law V play so narrowly circumscribed a role in Frege's proofs of his basic laws of arithmetic, it is straightforward to recast those proofs as derivations in Frege arithmetic.2 So the name 'Frege's theorem' is in order. Significant as he believed Frege's theorem to be, Boolos came not to praise Hume's principle but to cast doubt upon it. Crispin Wright had invoked the theorem to vindicate the epistemological kernel of Frege's logicism. Accord