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What's meaning of this symbol in set theory as following, which seems like $b$?

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I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma?

Thanks for any help:)

2 Answers 2

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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:

  1. $\aleph_0<\frak d\leq c$: 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 $\frak c$.

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

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

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    @Asaf: Somehow I suspected that Shelah was involved. :-) Thanks.2012-07-05
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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.