Eventual domination and the numbers b and d

Work in ZFC. Write ${}^{\omega }\omega$ for the set of functions $\omega \to \omega$ (Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation ${\kappa }^{\lambda }$, and why they are written apart from the ordinal operations), ordered pointwise; a function is thus a sequence of natural numbers (The natural numbers $\mathbb{N}$ (von Neumann)). For $f,g\in {}^{\omega }\omega$ put

$f{\le }^{\ast }g:⟺f(n)\le g(n)\text{ for all but finitely many }n\in \omega ,$

and say that $g$ eventually dominates $f$. The relation ${\le }^{\ast }$ is reflexive and transitive. A family $\mathcal{B}\subseteq {}^{\omega }\omega$ is

• ${\le }^{\ast }$-unbounded when there is no single $g\in {}^{\omega }\omega$ with $f{\le }^{\ast }g$ for every $f\in \mathcal{B}$;
• ${\le }^{\ast }$-dominating (equivalently ${\le }^{\ast }$-cofinal) when for every $f\in {}^{\omega }\omega$ there is $g\in \mathcal{B}$ with $f{\le }^{\ast }g$.

The bounding number. $b$ is the least cardinality of a ${\le }^{\ast }$-unbounded family $\mathcal{B}\subseteq {}^{\omega }\omega$.

The dominating number. $d$ is the least cardinality of a ${\le }^{\ast }$-dominating family $\mathcal{D}\subseteq {}^{\omega }\omega$.

Both minima exist and are cardinals. The collection of candidate cardinalities for $b$ is the image under $X\mapsto \mid X\mid$ of a subset of the power set of ${}^{\omega }\omega$, hence is a set of ordinals by Replacement, and it is nonempty because ${}^{\omega }\omega$ itself is ${\le }^{\ast }$-unbounded: given any $g$, the function $n\mapsto g(n)+1$ lies in ${}^{\omega }\omega$ and is not ${\le }^{\ast }$-below $g$. The same argument shows the candidates for $d$ form a nonempty set of ordinals, because ${}^{\omega }\omega$ is ${\le }^{\ast }$-dominating, each $f$ being dominated by itself. A nonempty set of ordinals has a least element, and that element is a cardinal by A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used; the Axiom of Choice (The Axiom of Choice) is what makes the cardinalities $\mid X\mid$ available. Thus

$b=\min \{\mid \mathcal{B}\mid :\mathcal{B}\subseteq {}^{\omega }\omega \text{ is }{\le }^{\ast }\text{-unbounded}\},d=\min \{\mid \mathcal{D}\mid :\mathcal{D}\subseteq {}^{\omega }\omega \text{ is }{\le }^{\ast }\text{-dominating}\},$

and the minima are attained, so there are an unbounded family of size $b$ and a dominating family of size $d$.

Conventions. The order on ${}^{\omega }\omega$ is the eventual one above; pointwise domination of a finite family is computed by pointwise maxima, which is the observation behind the elementary bounds proved in Basic bounding and dominating relations. Some sources write $f{<}^{\ast }g$ for "eventually strictly below" and define $b$ and $d$ with ${\le }^{\ast }$; the two readings give the same numbers, since replacing $g$ by $n\mapsto g(n)+1$ turns ${\le }^{\ast }$-domination into ${<}^{\ast }$-domination. Monk and Bartoszyński use exactly the definitions above, with $\mathfrak{c}=2^{{\mathrm{\aleph }}_0}$ as the largest candidate size.

The set ${}^{\omega }\omega$ has cardinality $\mathfrak{c}$ under AC, and ${\le }^{\ast }$ depends only on the eventual behaviour of a function; both facts are used in Basic bounding and dominating relations, where the chain ${\mathrm{\aleph }}_1\le b=\mathrm{cf}(b)\le \mathrm{cf}(d)\le d\le \mathfrak{c}$ is proved. Nothing in the definition requires that a dominating or unbounded family be closed under finite modifications: both properties are preserved when the family is enlarged, and replacing each member $f$ by its running maximum $n\mapsto {\max }_{m\le n}f(m)$ preserves both properties, since $f{\le }^{\ast }g$ implies $f{\le }^{\ast }{g}_{\max }$ and $f\le f_{\max }$ pointwise. A dominating or unbounded family may therefore be assumed to consist of nondecreasing functions.