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Transactions of the American Mathematical Society

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Peano arithmetic and hyper-Ramsey logic

Author: James H. Schmerl
Journal: Trans. Amer. Math. Soc. 296 (1986), 481-505
MSC: Primary 03H15; Secondary 03C80, 03C85, 03F35
MathSciNet review: 846594
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Abstract: It is known that $ {\text{PA}}({Q^2})$, Peano arithmetic in a language with the Ramsey quantifier, is complete and compact and that its first-order consequences are the same as those of $ \Pi _1^1{\text{-CA}_0}$. A logic $ \mathcal{H}{\mathcal{R}_\omega }$, called hyper-Ramsey logic, is defined; it is the union of an increasing sequence $ \mathcal{H}{\mathcal{R}_1} \subseteq {\mathcal{H}_{\mathcal{R}2}} \subseteq \mathcal{H}{\mathcal{R}_3} \subseteq \cdots $ of sublogics, and $ \mathcal{H}{\mathcal{R}_1}$ contains $ L({Q^2})$. It is proved that $ {\text{PA}}(\mathcal{H}{\mathcal{R}_n})$, which is Peano arithmetic in the context of $ \mathcal{H}{\mathcal{R}_n}$, has the same first-order consequences as $ \Pi _n^1{\text{-CA}_0}$. A by-product and ingredient of the proof is, for example, the existence of a model of $ {\text{CA}}$ having the form $ (\mathcal{N}, {\text{Class}}(\mathcal{N}))$.

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Keywords: Peano arithmetic, second-order theories of arithmetic, Ramsey quantifier, ramified analytical hierarchy, hyper-Ramsey logic
Article copyright: © Copyright 1986 American Mathematical Society

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