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

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.