# Mad families and non-meager filters

## TL;DR

This paper proves the consistency of the existence of a non-meager filter on ω alongside the non-existence of mad families within ZF+DC+ZFC, addressing a question by Neeman and Norwood.

## Contribution

It introduces a weaker form of madness and strengthens previous results by showing no mad families exist in the constructed model.

## Key findings

- Existence of a non-meager filter on ω is consistent with no mad families.
- A weaker version of madness is introduced and analyzed.
- The result strengthens previous work by eliminating mad families in the model.

## Abstract

We prove the consistency of ZF+DC+"there are no mad families"+"there exists a non-meager filter on $\omega$" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.