Frege's unofficial arithmetic

I show that any sentence of nth-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically, I set forth an enriched second-order language L, a sentence A of L (which is true on the intended interpretation of L), and a compositionally recursive transformation Tr defined on formulas of L, and show that they have the following two properties: (a) in a universe with at least beth(n-2) objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L. and (b) for any sentence [phi] of L [phi(Tr)] is a second-order sentence containing no arithmetical vocabulary, and A proves [phi <----> phi(Tr)] .