# Martin’s Maximum, saturated ideals, and nonregular ultrafilters. Part I.

### Abstract

The authors present a provably strongest form of Martin’s axiom, called Martin’s Maximum and show its consistency. From it we derive the solutions to several classical problems in set theory, showing that $2^{\mathfrak{N_0}} = \mathfrak{N}_2$, the non-stationary ideal on $\omega_1$ is $\mathfrak{N}_2$-saturated, and several other results. We show as a consequence of our techniques that there can be no “nice” inner model of a supercompact cardinal. We generalize our results to cardinals above $\omega_1$ to show, for example, the consistency of the statement “The non-stationary ideal on every regular cardinal $\kappa$ is precipitous.”

## Authors

Matthew Dean Foreman

Menachem Magidor

Saharon Shelah