The Journal of Symbolic Logic

Research Article

Consistency results about filters and the number of inequivalent growth types

Andreas Blassa1 and Claude Laflammea2

a1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

a2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada

We use models of set theory described in [2] and [3] to prove the consistency of several combinatorial principles, for example:

If is any filter on N containing all the cofinite sets, then there is a finite-to-one function f: N → N such that f() is either the filter of cofinite sets or an ultrafilter.

As a consequence of our combinatorial principles, we also obtain the consistency of:

The partial ordering P of slenderness classes of abelian groups, denned and studied in [4], is a four-element chain.

In the remainder of this Introduction, we shall define our terminology and state the combinatorial principles to be considered. In §2, we shall establish some implications between these principles. In §3, we shall prove our consistency results by showing that the strongest of our principles holds in models of set theory constructed in [2] and [3].

A filter on N will always mean a proper filter containing all cofinite sets; in particular, an ultrafilter will necessarily be nonprincipal. We write N ↗ N for the set of nondecreasing functions from the set N of positive integers into itself. A subset of N ↗ N is called an ideal if it is closed downward (if f(n) ≤ g(n) for all n and if g ∈ ℐ, then f ∈ ℐ) and closed under binary maximum (if f(n) = max(g(n), h(n)) for all n and if g, h ∈ ℐ then f ∈ ℐ).

(Received July 13 1987)