## Martin’s maximum, saturated ideals, and nonregular ultrafilters. I.(English)Zbl 0645.03028

Ann. Math. (2) 127, No. 1, 1-47 (1988); correction ibid. 129, No. 3, 651 (1989).
This is a long awaited and important paper. It contains a lot of information on the topics mentioned in the title, and it deserves to be read by everyone.
Call a poset P stationarity preserving if $$\Vdash_ P$$ “S is stationary” whenever $$S\subseteq \omega_ 1$$ is stationary. It is shown that if P is not stationarity preserving then there are $$\omega_ 1$$ dense subsets of P such that no filter on P meets them all. Thus the following statement is basically the best one can hope for when one tries to generalize Martin’s Axiom:
Martin’s Maximum: whenever P is a stationarity preserving poset and $${\mathcal D}$$ is a collection of $$\omega_ 1$$ dense sets in P there is a filter G on P intersecting every $$D\in {\mathcal D}.$$
The authors show Martin’s Maximum consistent using a supercompact cardinal. They then go on to point out several consequences of Martin’s Maximum such as: $$2^{\aleph_ 0}=\aleph_ 2$$ (which to this reviewer’s knowledge is not known to follow from the Proper Forcing Axiom), if $$\kappa \geq \omega_ 2$$ is regular and $$A\subseteq \{\delta <\kappa:$$ $$cf(\delta)=\omega \}$$ is stationary then A contains a closed copy of $$\omega_ 1$$, the nonstationary ideal on $$\omega_ 1$$ is $$\aleph_ 2$$-saturated (even after forcing with a ccc poset) and precipitous, and the Chang conjecture.
After a short section on versions of Martin’s Maximum which are consistent with CH the authors generalize their work to higher cardinals to obtain results on the precipitousness of nonstationary ideals. A remarkable result is the following: relative to a supercompact cardinal it is consistent that on every regular cardinal the nonstationary ideal is precipitous.
Remarks: 1. The authors show Martin’s Maximum consistent by showing the following consistent: the Semi-Proper Forcing Axiom plus “every stationarity preserving poset is $$\aleph_ 1$$-semi-proper”. In J. Symb. Logic 52, 360-367 (1987; Zbl 0625.03035), S. Shelah showed that the Semi-Proper Forcing Axiom actually implies Martin’s Maximum. Thus, as $$\aleph_ 1$$-semi-proper posets are stationarity preserving, Martin’s Maximum and the Semi-Proper Forcing Axiom are equivalent. 2. Part II - devoted mainly to ultrafilters - has since appeared: ibid. 127, 521-545 (1988).
Reviewer: E.Hartová

### MSC:

 03C55 Set-theoretic model theory 03E35 Consistency and independence results 03E50 Continuum hypothesis and Martin’s axiom 03E05 Other combinatorial set theory

Zbl 0625.03035
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