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A variant of Mathias forcing that preserves \({\mathsf{ACA}_0}\)

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Abstract

We present and analyze \({F_\sigma}\)-Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as \({\mathsf{ACA}_0}\) and \({\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}\), whereas Mathias forcing does not. We also show that the needed reals for \({F_\sigma}\)-Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing.

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  1. François G. Dorais

    Present address: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH, 03755, USA

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  1. Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI, 48109, USA

    François G. Dorais

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  1. François G. Dorais
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Correspondence to François G. Dorais.

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Dorais, F.G. A variant of Mathias forcing that preserves \({\mathsf{ACA}_0}\) . Arch. Math. Logic 51, 751–780 (2012). https://doi.org/10.1007/s00153-012-0297-4

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