Basic bounding and dominating relations

Every countable family is bounded: given $⟨f_n:n\in \mathbb{N}⟩$ in ${}^{\omega }\omega$, define $g(n)$ as the greatest element of the finite nonempty set $\{f_0(n),\dots ,f_n(n)\}$, which exists by [F5]; then for each $i$ and every $n\ge i$ one has $f_i(n)\le g(n)$, so $f_i{\le }^{\ast }g$. Hence no family of size at most ${\mathrm{\aleph }}_0$ is ${\le }^{\ast }$-unbounded, and since $b$ is a cardinal that is the least size of an unbounded family, $b>{\mathrm{\aleph }}_0$, that is, ${\mathrm{\aleph }}_1\le b$.

No countable family is dominating: the empty family is not dominating, and any nonempty finite or countably infinite family can be listed as $D=\{h_i:i\in \omega \}$, repeating entries if necessary. Set $q(n)=1+\max \{h_i(n):i\le n\}$, so for each $i$ one has $q(n)>h_i(n)$ whenever $n\ge i$. Thus $d\ge {\mathrm{\aleph }}_1$. An infinite cardinal is a limit ordinal, and [F2] gives $\mathrm{cf}(d)\le d$.

$d\le \mathfrak{c}$: the map $f\mapsto \{(n,f(n)):n\in \mathbb{N}\}$ injects ${}^{\omega }\omega$ into $\mathcal{P}(\omega \times \omega )$, so by [F4] $\mid {}^{\omega }\omega \mid \le 2^{\mid \omega \times \omega \mid }=2^{{\mathrm{\aleph }}_0\otimes {\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0}=\mathfrak{c};$ and ${}^{\omega }\omega$ is ${\le }^{\ast }$-dominating, since $f{\le }^{\ast }f$ for every $f$. Hence some dominating family has size at most $\mathfrak{c}$, and $d\le \mathfrak{c}$.

$b=\mathrm{cf}(b)$: by [F2] $\mathrm{cf}(b)\le b$, so suppose $\mathrm{cf}(b)<b$. By [F1] fix an unbounded family $\{f_{\xi }:\xi <b\}$ of size $b$, and by [F2] fix a strictly increasing cofinal map $\alpha \mapsto b_{\alpha }$ from $\lambda :=\mathrm{cf}(b)$ into $b$. For each $\alpha <\lambda$ the subfamily $B_{\alpha }=\{f_{\xi }:\xi <b_{\alpha }\}$ has cardinality at most $b_{\alpha }<b$, so by the minimality in [F1] it is bounded: choose $g_{\alpha }$ with $f_{\xi }{\le }^{\ast }{g}_{\alpha }$ for every $\xi <b_{\alpha }$ (the choices are made by [F3]). The family $\{g_{\alpha }:\alpha <\lambda \}$ has size at most $\lambda =\mathrm{cf}(b)<b$, so it too is bounded; fix $h$ with $g_{\alpha }{\le }^{\ast }h$ for every $\alpha <\lambda$. Every $\xi <b$ satisfies $\xi <b_{\alpha }$ for some $\alpha <\lambda$, because the map is cofinal, so $f_{\xi }{\le }^{\ast }{g}_{\alpha }{\le }^{\ast }h$ and $h$ bounds the allegedly unbounded family, a contradiction. Hence $\mathrm{cf}(b)=b$ and $b$ is regular.

$b\le \mathrm{cf}(d)$: let $D=\{h_{\xi }:\xi <d\}$ be a dominating family of size $d$ by [F1], put $\lambda =\mathrm{cf}(d)$, and fix a strictly increasing cofinal map $\alpha \mapsto d_{\alpha }$ from $\lambda$ into $d$ by [F2]. For $\alpha <\lambda$ put $D_{\alpha }=\{h_{\xi }:\xi <d_{\alpha }\}$; then $\mid D_{\alpha }\mid \le d_{\alpha }<d$, the sets $D_{\alpha }$ increase with $\alpha$, and ${\bigcup }_{\alpha <\lambda }{D}_{\alpha }=D$. No $D_{\alpha }$ is dominating, since $d$ is the least size of a dominating family, so by [F3] choose $f_{\alpha }\in {}^{\omega }\omega$ not dominated by any member of $D_{\alpha }$. The family $\{f_{\alpha }:\alpha <\lambda \}$ is unbounded: if some $g$ satisfied $f_{\alpha }{\le }^{\ast }g$ for every $\alpha$, then by domination some $h\in D$ satisfies $g{\le }^{\ast }h$, and $h\in D_{\alpha }$ for some $\alpha$, so $f_{\alpha }{\le }^{\ast }g{\le }^{\ast }h$ contradicts the choice of $f_{\alpha }$. Therefore $b\le \mid \{f_{\alpha }:\alpha <\lambda \}\mid \le \lambda =\mathrm{cf}(d)$.

Steps 1.1, 1.2, 1.4 and 1.5 give ${\mathrm{\aleph }}_1\le b=\mathrm{cf}(b)\le \mathrm{cf}(d)\le d$, and step 1.3 adds $d\le \mathfrak{c}=2^{{\mathrm{\aleph }}_0}$; together these are the displayed chain. This is the statement. ∎