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) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing.
Categories: math.LO