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**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).

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 |