Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

TL;DR

This paper explores the Borel Tukey ordering on continuum cardinal invariants, extending its applicability, constructing Borel maps for key inequalities, and embedding the inclusion order into this framework.

Contribution

It introduces a broader class of invariants for Borel Tukey ordering, constructs Borel maps for inequalities, and embeds the inclusion order into the Borel Tukey framework.

Findings

Borel Tukey ordering applies to a larger class of cardinals.

Constructed Borel Tukey maps for key inequalities like ndwenf6n's diagram.

Embedded the inclusion order on (f5) into the Borel Tukey ordering.

Abstract

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality $\mathfrak p\leq\mathfrak b$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we show that the inclusion ordering on $\mathcal P(\omega)$ embeds into the Borel Tukey ordering on cardinal invariants.