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

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 ${\mathsf{WKL}}_0+\mathsf{I}{\Sigma }_2^0$, whereas Mathias forcing does not. We also show that the needed reals for $F_{\sigma }$-Mathias forcing (in the sense of Blass) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing.

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