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Frege's unofficial arithmetic
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- Journal:
- The Journal of Symbolic Logic / Volume 67 / Issue 4 / December 2002
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1623-1638
- Print publication:
- December 2002
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- Article
- Export citation
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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 
of 
of 
is a second-order sentence containing no arithmetical vocabulary, and 
