What is the cardinality of the set $F$ of all normal functions $f \colon \omega_1 \to \omega_1$, where $\omega_1$ is the first uncountable ordinal? What is the least cardinality of a subset of $F$ such that every function in $F$ is bounded by some element of the subset?

For both questions, the answer does not change when you remove the word "normal" from the question. For the first question: There is a 11 map $f\mapsto N(f)$ that assigns to each function $f$ a normal function $N(f)$. $N(f)(\alpha)$ just adds up all values of $f$ below $\alpha$. (Or better: of $f+1$, to make it strictly increasing.) So there are $2^{\aleph_1}$ many of them. For the second question: Let $\mathfrak d(\kappa)$ be the smallest number functions needed to dominate all functions from $\kappa$ to $\kappa$. James Cummings and Saharon Shelah (Cardinal invariants above the continuum, Ann. Pure Appl. Logic 75 (1995), no. 3, 251–268) showed that, just like the continuum functions $\kappa\mapsto 2^\kappa$, also the "dominating" function $\kappa\mapsto \mathfrak d(\lambda)$ can have quite arbitrary behaviour. In particular, both $\mathfrak d(\aleph_1)=2^{\aleph_1}$ and $\mathfrak d(\aleph_1)< 2^{\aleph_1}$ are consistent. (This specific result for $\aleph_1$ may be older, though.) References:


