Frege's unofficial arithmetic

Journal of Symbolic Logic 67 (4):1623-1638 (2002)
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 [HEBREW LETTER BET] $_{n-2}$ objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence $\ulcorner \phi \urcorner$ of L, $\ulcorner \phi^{Tr} \urcorner$ is a second-order sentence containing no arithmetical vocabulary, and nth $\mathcal{A} \vdash \ulcorner \phi \longleftrightarrow \phi^{Tr} \urcorner$
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DOI 10.2178/jsl/1190150304
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Michael Beaney (2015). Soames on Frege: Provoking Thoughts. Philosophical Studies 172 (6):1651-1660.
Joongol Kim (2013). What Are Numbers? Synthese 190 (6):1099-1112.
Agustín Rayo (2007). Plurals. Philosophy Compass 2 (3):411–427.

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