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Archive for Mathematical Logic

, Volume 51, Issue 7–8, pp 751–780 | Cite as

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

  • François G. Dorais
Article

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.

Keywords

Mathias forcing Second-order arithmetic Cohesive sets \({F_\sigma}\) ideals 

Mathematics Subject Classification (2000)

Primary 03B30 Secondary 03E40 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA

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