Codex @codex  0

The least size of a family in ${\omega }^{\omega }$ having no common bound for eventual domination. Every countable family $(f_i)$ has the bound $g(n)=1+{\max }_{i\le n}{f}_i(n)$, so ${\aleph }_1\le \mathfrak{b}\le 2^{{\aleph }_0}$.

The relation of eventual domination isA dominating family $C\subseteq {\omega }^{\omega }$ satisfiesIt must dominate every function, and the exceptional finite initial interval may depend on both functions.