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

TL;DR

This paper introduces $F_\sigma$-Mathias forcing, a modified version of Mathias forcing that preserves weak subsystems of second-order arithmetic like $ ext{ extsf{ACA}}_0$, and shows it only requires computable reals.

Contribution

It defines and analyzes $F_\sigma$-Mathias forcing, demonstrating its preservation of certain subsystems and its reliance solely on computable reals, unlike traditional Mathias forcing.

Findings

Preserves $ ext{ extsf{ACA}}_0$ and $ ext{ extsf{WKL}}_0 + ext{ extsf{I}} ext{ extsf{ extSigma}}^0_2$

Requires only computable reals for needed reals

Is a tamer alternative to Mathias forcing

Abstract

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.