## Abstract

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)] .

Original language | English |
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Pages (from-to) | 1623-1638 |

Number of pages | 16 |

Journal | Journal of Symbolic Logic |

Volume | 67 |

Publication status | Published - Dec 2002 |

## Keywords

- second-order logic
- arithmetic
- logicism
- nominalism
- Frege
- Hodes
- Boolos