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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 
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 ℶn−2 objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence 
of 
is a second-order sentence containing no arithmetical vocabulary, and 
