In set theory, what does the symbol $\mathfrak b$ mean?

Question

In set theory, what does the symbol $\mathfrak b$ mean? Could somebody tell me something basic about $\mathfrak b$? In particulat, I want to know the relation between $\mathfrak b$ and $\mathfrak c$. Thanks ahead.

Answer 1

It is the bounding number. That's the smallest cardinality of a family $\mathcal F \subseteq \{f: \omega \to \omega \}$ such that $\mathcal F$ cannot be dominated that is, such that there is no function $g$ dominating all functions in $\mathcal F$.

A function $g$ is said to dominate $f$ if there is $n$ such that for $k > n$, $f(k) < g(k)$

Of course the set of all functions cannot be bounded hence $\mathfrak b \le \mathfrak c$.

Answer 2

It’s the bounding number:

$$\mathfrak b=\min\{|F|:F\subseteq{}^\omega\omega\text{ and }\forall g\in{}^\omega\omega\exists f\in F(g\le^* f)\}\;,$$

where $g\le^* f$ means $\{k\in\omega:f(k)<g(k)\}$ is finite. It’s not hard to prove that $\omega_1\le\mathfrak b\le 2^\omega$. The link includes information on the relationship between $\mathfrak b$ and some other numbers between $\omega_1$ and $2^\omega$.