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

##### By F. G. Dorais

###### Archive for Mathematical Logic 51 (2012), 751–780

- mr: 2975428
- zbl: 1269.03020
- arxiv: 1110.6559
- doi: 10.1007/s00153-012-0297-4

We present and analyze -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 and , whereas Mathias forcing does not. We also show that the needed reals for -Mathias forcing (in the sense of Blass) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing.