A Lower Bound for Generalized Dominating Numbers

Abstract

We show a new proof for the fact that when $κ$ and $λ$ are infinite cardinals satisfying $λ^ κ= λ$, the cofinality of the set of all functions from $λ$ to $κ$ ordered by everywhere domination is $2^λ$. An earlier proof was a consequence of a result about independent families of functions. The new proof follows directly from the main theorem we present: for every $A \subseteq λ$ there is a function $f: {^κλ} \to κ$ such that whenever $M$ is a transitive model of $\textrm{ZF}$ such that ${^κλ} \subseteq M$ and some $g: {^κλ} \to κ$ in $M$ dominates $f$, then $A \in M$. That is, "constructibility can be reduced to domination".