# The covering number of the strong measure zero ideal can be above almost everything else

## TL;DR

This paper demonstrates that certain tree forcings, like Sacks forcing, can increase the covering number of the strong measure zero ideal beyond most classical continuum invariants, with new consistency results on their relationships.

## Contribution

It shows that Sacks forcing can make the covering number of the strong measure zero ideal equal to the continuum and separate multiple cardinal invariants, a novel consistency result.

## Key findings

- Covering number of $ ext{SN}$ can equal the continuum in Sacks model.
- $ ext{non}( ext{SN})< ext{cov}( ext{SN})< ext{cof}( ext{SN})$ can hold simultaneously.
- $ ext{SN} ext{ is contained in } s^0$ in ZFC.

## Abstract

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal{SN}$ are pairwise different. Another consequence is that $\mathcal{SN}\subseteq s^0$ in ZFC where $s^0$ denotes the Marczewski's ideal.