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

We present and analyze $F_σ$-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 $\mathsf{WKL}_0 + \mathsf{I}Σ^0_2$, whereas Mathias forcing does not. We also show that the needed reals for $F_σ$-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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