Learning project #1: Foreman-Magidor-Shelah "Martin's Maximum, saturated ideals, and nonregular ultrafilters I"

The paper "Martin's Maximum, saturated ideals, and nonregular ultrafilters I" by Foreman-Magidor-Shelah seems to be perhaps the most influential set theory paper in recent times (let's use a loose definition of "recent"). In this paper, they introduce the "maximal" forcing axiom known as Martin's Maximum, and then prove some consistency results from large cardinals about the saturation of "natural" ideals. We will focus on the second thing; for example:

1. Forcing with $\rm{Col}(\mu,<\kappa)$ makes the nonstationary ideal restricted to $[\mu]^{<\mu}$ precipitous.

2. Forcing with  $\rm{Col}(\omega_1,<\kappa)$ gives an $\aleph_2$-saturated ideal on $\omega_1$.

3. To make the nonstationary ideal on $\omega_1$ saturated, we must force with something more than the collapse. But this is also possible and can be done with $\kappa$-c.c. semiproper forcing.

All of the conclusions above are also consequences of Martin's Maximum.

Recall the definition and motivation of precipitous ideals from the beginning of a previous post. In that post, we gave the Jech--Magidor--Mitchell--Prikry construction of a precipitous ideal on $\omega_1$. These authors in fact proved that the nonstationary ideal on $\omega_1$ can be precipitous. The method was by an iterated forcing that at each step destroyed the stationarity of sets in the ideal constructed so far.

However, the FMS paper managed to find a different consistency proof of precipitousness of the nonstationary ideal on $\omega_1$. In fact, the proof is general to any other regular cardinals, unlike the JMMP construction which uses something special about $\omega_1$ for destroying stationarity (although generalizations of that construction to other cardinals has been carried out in work of Gitik). The drawback is that the FMS construction uses quite large cardinals, whereas Gitik found the equiconsistency.

I plan to at least cover (1.) and (2.) in a forthcoming series of posts. The plan is to prove (3.) sometime in a future learning project dealing with semiproper forcing, RCS iterations, and the other parts of the FMS paper.

The posts will come out slowly because I want to make sure I fully understand each step. The references will be the FMS paper and Section 8 of Foreman's Handbook chapter.