James E. Baumgartner

Applications of the Proper Forcing Axiom

Publisher Summary The chapter presents a more powerful version of Martin Axiom (MA), the Proper Forcing Axiom (PFA) that is successful at settling problems left open by MA. PFA is obtained from MA by replacing the countable chain condition by properness. In addition to the countable chain condition (ccc) partial orderings, the proper partial orderings include the countably closed and Axiom A orderings and many others as well. Like the ccc orderings, the proper orderings are closed under forcing composition. Thus, PFA applies to many more orderings than MA, and this is one source of its power. Another source of power is the fact that large cardinals are needed to obtain the consistency of PFA. This means that PFA can be applied to problems that MA has no hope of solving simply because of questions of consistency strength.

A new class of order types

Abstract Let φ 4 be the class of all order-types ϕ with the properties that every uncountable subtype of ϕ contains an uncountable well-ordering, but ϕ is not the union of countably many well-orderings. It is proved that φ 4 ≠ 0, and a way is found of associating stationary sets with most of the types in φ 4 which is useful for applications. A number of results concerning the structure and embeddability properties of φ 4 are obtained, including some consistency and independence results. One consequence is the independence of Jensens combinatorial principle □ ω 1 .

Ineffability Properties of Cardinals II

This paper applies the methods of [1] to several classes of cardinals other than ineffables. The central point is the same as [1], namely that many ‘large cardinal’ properties are better viewed as properties of normal ideals than as properties of cardinals alone, and that in order to understand these properties fully it is necessary to consider the associated normal ideals.

Adjoining dominating functions

If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number (and hence the distributivity number ) remains at ω 1 .

PARTITION THEOREMS AND ULTRAFILTERS

We introduce a class of ultrafilters on u called fc-arrow ultrafil- ters and characterized by the partition relation U -> ( U, k)2. These are studied in conjunction with P-points, Q-points, weakly Ramsey and Ramsey ultrafilters.

Sacks forcing and the total failure of Martin's Axiom

Abstract Using side-by-side Sacks forcing, it is proved relatively consistent that the continuum is large and Martins Axiom fails totally, that is, every c.c.c. space is the union of ℵ1 nowhere dense sets (equivelently, if P is a nontrivial partial ordering with the countable chain condition, then there are ℵ1 dense sets in P such that no filter in P meets them all).

On the size of closed unbounded sets

Abstract We study various aspects of the size, including the cardinality, of closed unbounded subsets of [λ]

Partition algebras for almost-disjoint families

A set a C X is a partitioner of a maximal almost-disjoint faculty F of subsets of X if every element of F is almost contained in or almost-disjoint from a. The partition algebra of F is the quotient of the Boolean algebra of partitioners modulo the ideal generated by F and the finite sets. We show that every countable algebra is a partition algebra, and that CH implies every algebra of cardinality < 28o is a partition algebra. We also obtain consistency and independence results about the representability of Boolean algebras as partition algebras. 0. Introduction. Two subsets a and b of w are almost-disjoint if a n b is finite. An almost-disjoint family, for the purposes of this paper, is an infinite set of infinite, pairwise almost-disjoint subsets of c. We shall be interested in maximal almost-disjoint families (called mad families by Mathias in [7]), which in particular must be uncountable. Let F be a mad family. A set a c w is a partitioner of F iff Vb E F either b a or b n a is finite. The set of partitioners of F forms a Boolean algebra B. A partitioner of F is nontrivial if it does not belong to the ideal I in B generated by F together with the finite sets. The algebra B/I is called the partition algebra of F. If a Boolean algebra is isomorphic to the partition algebra of some mad family then we say that the algebra is representable. It is our hope that this notion of partition algebra, due to the second author, will provide a useful way to classify almost-disjoint families. Other methods of classification have been studied in [3 and 9]. We are primarily concerned with determining when a given Boolean algebra is representable. We begin in ?1 with some technical results which yield, in ?2, the fact that every countable Boolean algebra is representable, and that certain uncountable algebras are always representable as well. We also show here that if B is a Boolean algebra and I B I is smaller than the Rothberger cardinal p, then B is representable. Hence, in particular, Martins Axiom implies that every Boolean algebra of cardinality < 28o is representable. We show in ?3 that the continuum hypothesis (CH) implies that every Boolean algebra of cardinality < 28o is representable, and in ?4 we show it consistent that CH fails and no algebra containing the free algebra on 8 2 generators is representable. Hence, in particular, it is consistent that no infinite Received by the editors April 29, 1981. 1980 Mathematics Subject Classification. Primary 03E50; Secondary 03E05, 06E05.

Ultrafilters on Ω

We study the I -ultrafilters on ω , where I is a a collection of subsets of a set X , usually ℝ or ω 1 . The I -ultrafilters usually contain the P -points, often as a small proper subset. We study relations between I -ultrafilters for various I , and closure of I -ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I -ultrafilters always exist.

Partition relations for countable topological spaces

Abstract We consider partition relations for pairs of elements of a countable topological space. For spaces with infinitely many nonempty derivatives a strong negative theorem is obtained. For example, it is possible to partition the pairs of rationals into countably many pieces so that every homeomorph of the rationals contains a pair from every piece. Some positive results are also proved for ordinal spaces of the form ωα + 1, where α is countable.