A Lower Bound for Generalized Dominating Numbers

Abstract

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 $f:{}^{\kappa }\lambda \to \kappa$ such that whenever $M$ is a transitive model of $\text{ZF}$ such that ${}^{\kappa }\lambda \subseteq M$ and some $g:{}^{\kappa }\lambda \to \kappa$ in $M$ dominates $f$, then $A\in M$. That is, "constructibility can be reduced to domination".

arXiv comment9 pages

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