# François G. Dorais

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

##### By F. G. Dorais
###### Archive for Mathematical Logic 51 (2012), 751–780

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.