CUNY Logic Workshop, New York, November 2011

Description: The talk is loosely about the so-called cardinal invariants of the continuum. This particular story begins with Vojtáš, who, in addition to having several strange accents in his name, realized that many cardinal invariants admit a natural definition of the form:

\[\min\set{\abs{F} \mid \forall x\;\exists y\in F\; xRy}\]

Here, $R$ is some binary relation, $x$ ranges over the domain of $R$, and $F$ ranges over subsets of the codomain of $R$. For instance, the so-called dominating number $\mathfrak{d}$ is defined by the relation $\leq^*$ (eventual domination) on $\omega^\omega$.

If $R$ and $R’$ are two relations, then $R$ is said to be above $R’$ in the Tukey order iff there exist maps $\phi$ from the domain of $R’$ to the domain of $R$ and $\psi$ from the codomain of $R$ to the codomain of $R’$ such that

\[\phi(x) R y \implies x R’ \psi(y)\]

The Tukey ordering is important because it corresponds very closely with inequality of the associated cardinal invariants. However, in practice one is more concerned with true inequalities, that is, inequalities which hold in all models of ZFC. For this reason, Blass proposed that we consider the Borel Tukey order, which is defined as above except that now $R$ and $R’$ are assumed to be relations defined on standard Borel spaces and the maps $\phi$ and $\psi$ are required to be Borel. The Borel Tukey order is known to have applications of a combinatorial nature in areas such as parameterized diamond principles and Borel equivalence relations.

In this talk, I will build upon some work of Mildenberger on the Borel Tukey ordering for a family of unsplitting relations. More generally, we will discuss the similarities and differences between the usual ordering and the Borel Tukey orderings on a modest collection of classical combinatorial cardinal invariants. For this, we will need to widen our attention slightly to cardinals which admit a definition of the form

\[\min\set{\abs{F} : P(F)\;\&\;\forall x\;\exists y\in F\;\; xRy}\]

where $R$ is as above and $P$ is some second order property of the families $F$. This will allow us to consider the Borel Tukey order on many more cardinals. For instance, we shall be able to handle the pseudo-intersection number $\mathfrak p$. This is defined to be the least cardinality of a family $F$ such that $F$ is centered and $\forall x\in[\omega]^\omega\;\exists y\in F$ such that $x\not\mathrel{\subset^*}y$.

This is joint work with Juris Steprāns and Tamás Mátrai.