Ramified Frege Arithmetic

Frege’s definitions of zero, predecession, and natural number will be explained below. As for second-order Dedekind-Peano arithmetic, the axiomatization most convenient for our purposes is the following: (1) N0 (2) Nx∧Pxy→ Ny (3) ∀x∀y∀z(Nx∧Pxy∧Pxz→ y = z) (4) ∀x∀y∀z(Nx∧Ny∧Pxz∧Pyz→ x = y) (5) ¬∃x(Nx∧Px0) (6) ∀x(Nx→∃y(Pxy)) (7) ∀F(F0∧∀x∀y(Nx∧Fx∧Pxy→ Fy)→∀x(Nx→ Fx) If (slightly non-standardly) we take Frege arithmetic to be the second-order theory whose non-logical axioms are HP and Frege’s definitions of the arithmetical notions, re-construed as axioms, then Frege’s Theorem may