What is the cardinality of the set $F$ of all [normal functions](https://en.wikipedia.org/wiki/Normal_function) $f \colon \omega\_1 \to \omega\_1$, 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?



Cardinality of cofinal set of normal functions $f \colon \omega_1 \to \omega_1$
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 $\kappa$.


* 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](https://arxiv.org/abs/math/9509228), [MR1355135](https://mathscinet.ams.org/mathscinet-getitem?mr=1355135)


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.)


