Partially conservative extensions of arithmetic

Author:
D. Guaspari

Journal:
Trans. Amer. Math. Soc. **254** (1979), 47-68

MSC:
Primary 03F30; Secondary 03F25, 03F40, 03H15

MathSciNet review:
539907

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Abstract: Let *T* be a consistent r.e. extension of Peano arithmetic; , the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting ``for free"); and , variables through the set of all classes , . The principal concern of this paper is the question: When can we find an independent sentence which is -conservative in the following sense: Any sentence in which is provable from is already provable from *T*? (Additional embellishments: Ensure that is not provably equivalent to a sentence in any class ``simpler'' than ; that is not conservative for classes ``more complicated'' than .) The answer, roughly, is that one can find such a , embellishments and all, unless and are so related that such a *obviously* cannot exist. This theorem has applications to the theory of interpretations, since `` is -conservative'' is closely related to the property `` is interpretable in *T*"-or to variants of it, depending on . Finally, we provide simple model theoretic characterizations of -conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (*T* then being an extension of *ZF*).

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DOI:
https://doi.org/10.1090/S0002-9947-1979-0539907-7

Article copyright:
© Copyright 1979
American Mathematical Society