Archive for Mathematical Logic 51 (7-8):751-780 (2012)

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
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DOI 10.1007/s00153-012-0297-4
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References found in this work BETA

On the Strength of Ramsey's Theorem.David Seetapun & Theodore A. Slaman - 1995 - Notre Dame Journal of Formal Logic 36 (4):570-582.
Ramsey's Theorem and Cone Avoidance.Damir Dzhafarov & Carl Jockusch Jr - 2009 - Journal of Symbolic Logic 74 (2):557 - 578.
A New Proof That Analytic Sets Are Ramsey.Erik Ellentuck - 1974 - Journal of Symbolic Logic 39 (1):163-165.

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