# Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II

@article{Foreman1988MartinsMS, title={Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II}, author={Matthew D. Foreman and Menachem Magidor and Saharon Shelah}, journal={Annals of Mathematics}, year={1988}, volume={127}, pages={1-47} }

We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of small cardinality. Part II is logically independent of Part I.

## 307 Citations

### (kappa,theta)-weak normality

- Mathematics
- 2011

We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.

### Almost-disjoint coding and strongly saturated ideals

- Mathematics
- 2005

We show that Martin's Axiom plus c = N 2 implies that there is no (N 2 , N 2 , N 0 )-saturated cr-ideal on ω 1 .

### An ℵ1-dense ideal on ℵ2

- Mathematics
- 1998

This paper establishes the consistency of a countably complete, uniform, ℵ1-dense ideal on ℵ2. As a corollary, it is consistent that there exists a uniform ultrafilterD on ω2 such that |ω1ω2D|=ω1. A…

### Non-regular ultrafilters

- Mathematics
- 1994

We construct non-regular ultrafilters, extending filters which are dual to dense or layered ideals.

### The nonstationary ideal and the other -ideals on ₁

- Mathematics
- 2000

Under Martin’s Maximum every σ-ideal on ω1 is a subset of an ideal Rudin-Keisler reducible to a finite Fubini power of the nonstationary ideal restricted to a positive set.

### Proper forcings and absoluteness in L ( R )

- Mathematics
- 2010

We show that in the presence of large cardinals proper forcings do not change the theory of L(R) with real and ordinal parameters and do not code any set of ordinals into the reals unless that set…

### Large cardinals imply that every reasonably definable set of reals is lebesgue measurable

- Mathematics, Economics
- 1990

We prove that if there is a supercompact cardinal or much smaller large cardinals, then every set of reals from L(R) is Lebesgue measurable, and similar results. We also introduce some large…

### Perfect-Set Properties inL(R)[U]☆

- Mathematics, Economics
- 1998

Abstract We study perfect-set properties in the modelL( R )[U] whenL( R ) is (elementarily equivalent to) a Solovay model andUis a selective ultrafilter on the integers, generic overL( R ).

### The nonstationary ideal in the ℙmax extension

- MathematicsJournal of Symbolic Logic
- 2007

The Boolean algebra induced by the nonstationary ideal on ω1 in this model is studied and it is shown that the induced quotient does not have a simply definable form.

## References

SHOWING 1-10 OF 21 REFERENCES

### Separating ultrafilters on uncountable cardinals

- Mathematics
- 1984

A uniform ultrafilterU on κ is said to be λ-separating if distinct elements of the ultrapower never projectU to the same uniform ultrafilterV on λ. It is shown that, in the presence of CH, an…

### Precipitous ideals and Σ14 setssets

- Mathematics
- 1980

AbstractWe prove under the assumption of the existence of a measurable, cardinal and precipitous ideal onw1 that every Σ13 set is Lebesgue measurable, has the Baire property and is either countable…

### A model of set-theory in which every set of reals is Lebesgue measurable*

- Economics, Mathematics
- 1970

We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF,…

### Making the supercompactness of κ indestructible under κ-directed closed forcing

- Mathematics
- 1978

A model is found in which there is a supercompact cardinal κ which remains supercompact in any κ-directed closed forcing extension.

### On the singular cardinals problem I

- Mathematics
- 1977

We show how to get a model of set theory in which ℵω is a strong limit cardinal which violates the generalized continuum hypothesis. Generalizations to other cardinals are also given.

### Chang's conjecture and powers of singular cardinals

- MathematicsJournal of Symbolic Logic
- 1977

It is shown that the same conclusion can be derived from Chang's Conjecture which is, at least consistencywise, a weaker assumption than the existence of an ω 2 saturated ideal on ω 1 .

### Weakly normal filters and irregular ultrafilters

- Mathematics
- 1976

For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of…

### On the Singular Cardinals Problem

- Mathematics
- 1981

In this paper we show, for example, that if the GCH holds for every cardinal less than tc, a singular cardinal of uncountable cofinality, then the GCH holds at tc itself. This result is contrary to…

### Iterated forcing and changing cofinalities

- Mathematics
- 1981

AbstractWe weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the…