In set theory, what does the symbol $\mathfrak d$ mean? What's meaning of this symbol in set theory as following, which seems like $b$?   



I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma?
Thanks for any help:)
 A: It is the German script $\mathfrak{d}$ given by the LaTeX \mathfrak{d}. It probably represents a cardinal number (sometimes $\mathfrak{c}$ is used to represent the cardinality of the real numbers), but it would definitely depend on the context of what you are reading.
A: The symbol $\mathfrak d$ is used to denote the dominating number of the continuum.
If $g,f\colon\omega\to\omega$ we say that $g$ dominates $f$ if for all but finitely many $n$, $f(n)\leq g(n)$.
The dominating number is the smallest cardinality of a dominating family, namely the minimal $|F|$ such that $F\subseteq\omega^\omega$ and for every $f\colon\omega\to\omega$ there is some $g\in F$ such that $g$ dominates $f$.
Some observations:


*

*$\aleph_0<\frak d\leq c$: the former is true because if we have a countable family of functions by diagonalization argument we can produce a non-dominated function; the latter is true because it is obvious that $F=\omega^\omega$ is a dominating family and its size is exactly $\frak c$.

*If $\aleph_1=\frak c$ then $\frak c=d$, which is a trivial consequence of the above.

*It is not provable that there is an equality, because by forcing we can ensure that $\frak d<\frak c$.
