Finitude and Hume's principle

Journal of Philosophical Logic 26 (6):589-617 (1997)
Abstract
The paper formulates and proves a strengthening of Freges Theorem, which states that axioms for second-order arithmetic are derivable in second-order logic from Humes Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. Finite Humes Principle also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Freges definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.
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Richard Heck (2011). Ramified Frege Arithmetic. Journal of Philosophical Logic 40 (6):715-735.

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