TITLE: How many orders of infinity are there?
QUESTION [84 upvotes]: Define a growth function to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions.  Let's say that one growth function $F$ dominates another $G$ if one has $F(n) \geq G(n)$ for all $n$.  (One could instead ask for eventual domination, in which one works with sufficiently large $n$ only, or asymptotic domination, in which one allows a multiplicative constant $C$, but it seems the answers to the questions below are basically the same in both cases, so I'll stick with the simpler formulation.)  
Let's call a collection ${\mathcal F}$ of growth functions complete cofinal if every growth function is dominated by at least one growth function in ${\mathcal F}$.  
Cantor's diagonalisation argument tells us that a cofinal set of growth functions cannot be countable.  On the other hand, the set of all growth functions has the cardinality of the continuum.  So, on the continuum hypothesis, a cofinal set of growth functions must necessarily have the cardinality of the continuum.
My first question is: what happens without the continuum hypothesis?  Is it possible to have a cofinal set of growth functions of intermediate cardinality?
My second question is more vague: is there some simpler way to view the poset of growth functions under domination (or asymptotic domination) that makes it easier to answer questions like this?  Ideally I would like to "control" this poset in some sense by some other, better understood object (e.g. the first uncountable ordinal, the nonstandard natural numbers, or the Stone-Cech compactification of the natural numbers).
EDIT: notation updated in view of responses.

REPLY [4 votes]: I would like to make two points addressing the topological aspect of the OP by Terence Tao. The cardinals $\frak b$ and $\frak d$, although defined "arithmetically," are in fact equivalent to some incompleteness parameters of the Boolean Algebra $P (\omega) / fin$, equivalently of the space $\omega^* = \beta\omega \setminus \omega$.
1A) Indeed: $\frak b$ is the smallest cardinal of a family $\mathcal B$ of clopen subsets of $\omega^*$ such that, for some countable family $\mathcal A$ "disjointed" from $\mathcal B$ (and this means that for all $A$ in $\mathcal A$ and all $B$ in $\mathcal B$ the meet $A \wedge B = 0$), $\mathcal A$ and $\mathcal B$ cannot be separated by a clopen set.
This is well-defined because $P (\omega) / fin$ is not countably complete, i.e. countable subsets need not have the least upper bound.
On the other hand, it is well-known that  $P (\omega) / fin$ possesses the so called the du Bois-Reymond separation property (this means that disjointed  $\mathcal B$, $\mathcal A$ can be separated when both families are countable). A compact space with this property is called an $F$-space (even if it is  not Boolean, i.e. zero-dimensional), and the Boolean Algebra is called (by Eric van Douwen) weakly countably complete. 
1B) Now, regarding the cardinal $\frak d$, which is our main concern: $\frak d$ is the smallest cardinal for a $\mathcal B$ so that, for some countable $\mathcal A$ disjointed from $\mathcal B$, $\mathcal A$ and $\mathcal B$  cannot be "weakly left-separated." This means that there is no clopen set $C$ such that $\forall \  A \in \mathcal A$, $A \leq C$ and $\forall \  B \in \mathcal B$, $B - C \neq 0$.
These facts are, practically, obvious (so that I could not find a formal reference even in the excellent monograph of Andreas Blass he mentions in his answer).
2) My second remark is that the $\frak b$ and $\frak d$ so re-defined for  $P (\omega) / fin$  are (despite their "arithmetical" origin) perfectly applicable to any Boolean Algebra/Boolean space which is not countably complete, and that they measure the rough cardinal degree of such an incompleteness. 
(3) I also have some mild puzzlement about the a-symmetricity in the (revised) definition of $\frak d$.