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On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic

Author

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  • Vladimir Kanovei

    (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
    These authors contributed equally to this work.)

  • Vassily Lyubetsky

    (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either (1) the parameter-free countable axiom of choice AC ω * fails, or (2) AC ω * holds but the full countable axiom of choice AC ω fails in the domain of reals. In another generic extension of L , we define a set X ⊆ P ( ω ) , which is a model of the parameter-free part PA 2 * of the 2nd order Peano arithmetic PA 2 , in which CA ( Σ 2 1 ) (Comprehension for Σ 2 1 formulas with parameters) holds, yet an instance of Comprehension CA for a more complex formula fails. Treating the iterated Sacks forcing as a class forcing over L ω 1 , we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic PA 2 is formally consistent then so are the theories: (1) PA 2 + ¬ AC ω * , (2) PA 2 + AC ω * + ¬ AC ω , (3) PA 2 * + CA ( Σ 2 1 ) + ¬ CA .

Suggested Citation

  • Vladimir Kanovei & Vassily Lyubetsky, 2023. "On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic," Mathematics, MDPI, vol. 11(3), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:726-:d:1053554
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    References listed on IDEAS

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    1. Vladimir Kanovei & Vassily Lyubetsky, 2020. "Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level," Mathematics, MDPI, vol. 8(6), pages 1-46, June.
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