Cardinality of cofinal set of normal functions $f \colon \omega_1 \to \omega_1$ 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?
 A: For both questions, the answer does not change when you remove the word "normal" from the question.
For the first question: There is a 1-1 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 $\kappa$.

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*James Cummings and Saharon Shelah, Cardinal invariants above the continuum, Ann. Pure Appl. Logic 75 (1995), no. 3, 251–268, https://doi.org/10.1016/0168-0072(95)00003-Y,  arXiv:math/9509228, MR1355135
shows 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.)
