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Equivalence of families of functions on the natural numbers. (English) Zbl 0759.03025

The authors consider the following orderings on families of nondecreasing functions on the natural numbers (generalizing dominatedness by adding an accellerating parameter acting first on the range and second on the domain).
For \({\mathcal H},{\mathcal Y}\subseteq\omega\nearrow\omega\) they define \[ {\mathcal H}\precsim^ 1{\mathcal Y}\text{ iff }(\exists r\in\omega\nearrow\omega)(\forall f\in{\mathcal H})(\exists g\in{\mathcal Y})(\forall^ \infty_ n)(r(g(n))\geq f(n)), \]
\[ {\mathcal H}\precsim^ 2{\mathcal Y}\text{ iff }(\exists r\in\omega\nearrow\omega)(\forall f\in{\mathcal H})(\exists g\in{\mathcal Y})(\forall^ \infty_ n)(g(r(n))\geq f(n)). \] It is proved that under \({\mathfrak u}<{\mathfrak g}\) there are exactly 5 classes of families of functions for either ordering. Moreover, existence of exactly 5 classes of functions collapses the structure of types of ultrafilters (or subsets of \([\omega]^ \omega\)) under finite-to-one mappings. Consequences on classifications of measure zero sets are mentioned.

MSC:

03E35 Consistency and independence results
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E05 Other combinatorial set theory
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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