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# Consistency results about filters and the number of inequivalent growth types

Abstract

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).

References
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[1]Blass A., Near coherence of filters. II: Applications to operator ideals, the Stone-Čech remainder of the half-line, order ideals of sequences, and slenderness of groups, Transactions of the American Mathematical Society, vol. 300(1987), pp. 557581.
[2]Blass A. and Shelah S., Near coherence of filters. III: A simplified consistency proof (to appear).
[3]Blass A. and Shelah S., There may be simple and , points, and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213243.
[4]Göbel R. and Wald B., Wachstumstypen und schlanke Gruppen, Symposia Mathematica (INDAM), vol. 23, Academic Press, London, 1979, pp. 201239.
[5]Göbel R. and Wald B., Martin's axiom implies the existence of certain slender groups, Mathematische Zeitschrift, vol. 172(1980), pp. 107121.
[6]Shelah S., On cardinal invariants of the continuum, Axiomatic set theory, Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 183207.
[7]Specker E., Additive Gruppen von Folgen ganzer Zahlen, Portugaliae Mathematica, vol. 9 (1950), pp. 131140.
[8]van Douwen E. K., The integers and topology, Handbook of set-theoretic topology (Kunen K. and Vaughan J. E., editors), North-Holland, Amsterdam, 1984, pp. 111167.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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