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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:)

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2 Answers 2

up vote 13 down vote accepted

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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1  
What's the relation between $\mathfrak{d}$ and the $continuum$? –  Paul Jul 5 '12 at 12:16
    
You might want to see Cichoń's diagram (Wikipedia seems to be down now, but you can probably Google a backup or some other source). –  tomasz Jul 5 '12 at 13:48
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A good basic reference for $\mathfrak a,\mathfrak b,\mathfrak d,\mathfrak p,\mathfrak s$, and $\mathfrak t$ is Eric K. van Douwen, The Integers and Topology, in the Handbook of Set-Theoretic Topology, K. Kunen & J.E. Vaughan, eds. –  Brian M. Scott Jul 5 '12 at 19:00
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@Brian: There are circling rumors (with very credible sources) that $\frak p=t$ was proved recently. –  Asaf Karagila Jul 5 '12 at 21:49
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@Brian: Shelah and one of his model theory related postdoc, Malliaris. I heard the proof is related to their other work about regular ultrafilters. –  Asaf Karagila Jul 5 '12 at 21:53

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.

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