A Lower Bound for Generalized Dominating Numbers

We show a new proof for the fact that when $\kappa$ and $\lambda$ are infinite cardinals satisfying ${\lambda }^{\kappa }=\lambda$, the cofinality of the set of all functions from $\lambda$ to $\kappa$ ordered by everywhere domination is $2^{\lambda }$. 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 $A\subseteq \lambda$ there is a function $f: {^\kappa \lambda} \to \kappa$suchthatwhenever$M$isatransitivemodelof$$\text{ZF}$$suchthat${^\kappa \lambda} \subseteq M$andsome$g: {^\kappa \lambda} \to \kappa$in$M$dominates$f$,then$A \in M$. That is, “constructibility can be reduced to domination”.