The starting point of this question is the observation that in ${\sf (ZFC)}$, all ordinals $\alpha < \omega\_1$ can be [order-embedded in](https://math.stackexchange.com/questions/408300/countable-ordinals-are-embeddable-in-the-rationals-bbb-q-proofs-and-their) $\mathbb{R}$.


Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Given $f, g:\omega\to\omega$ we say $f\leq^\* g$ iff there is $N\in\omega$ such that $f(k) \leq g(k)$ for all $k\geq N$.


We say ${\cal D}\subseteq \omega^\omega$ is *dominating* if for all $f\in \omega^\omega$ there is $d\in {\cal D}$ such that $f\leq^\* d$, and we say ${\cal B}\subseteq \omega^\omega$ is *unbounded* if for all $f\in\omega^\omega$ there is $b\in {\cal B}$ such that $b\not\leq^\*f$. (A diagonalization argument shows that every unbounded (and therefore also every dominating) family must be uncountable, and there are interesting set-theoretical considerations in this context, see [here](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum).)


We define ${\frak b}$ to be the smallest cardinality that an unbounded family ${\cal B}\subseteq\omega^\omega$ can have, and let ${\frak d}$ be the smallest cardinality that a dominating family ${\cal D}\subseteq\omega^\omega$ can have. It is consistent that $\omega\_1 < {\frak b} < {\frak d} \leq 2^{\aleph\_0}$.


**Question.** Is it provable in ${\sf (ZFC)}$ that ${\frak b}$ or even ${\frak d}$ can be order-embedded in $\mathbb{R}$? (The question could also be asked of other [cardinal characteristics of the continuum](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum).)



Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$
Assume that $i : \alpha \to \mathbb{R}$ is an order-embedding of some ordinal $\alpha$ into $(\mathbb{R},<)$. We can modify $i$ to yield an order embedding $j : \alpha \to \mathbb{Q}$ by induction over $\beta < \alpha:


The interval $[i(\beta), i(\beta + 1)]$ is a non-degenerate interval in $\mathbb{R}$, which contains no points from the range of $i$ other than its endpoints. Pick rationals $p,q$ with $i(\beta) \leq p < q \leq i(\beta + 1)$, and set $j(\beta) = p$ (if $\beta$ is a limit ordinal, otherwise $j(\beta)$ is already defined) and $j(\beta + 1) = q$.


Thus, the ordinals that order-embed into $\mathbb{R}$ are precisely the countable ones.


