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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 $\frak d<\frak c$.

  • 1
    What's the relation between $\mathfrak{d}$ and the $continuum$?2012-07-05
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    You might want to see [Cichoń's diagram](http://en.wikipedia.org/wiki/Cicho%C5%84%27s_diagram) (Wikipedia seems to be down now, but you can probably Google a backup or some other source).2012-07-05
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    @John: Exactly what "the continuum" means seems to vary quite a bit with the occasion. Sometimes it is $\mathbb R$. Sometimes it is $\mathcal P(\omega)$ or $\mathbb 2^\omega$. Sometimes it is the [Baire space](http://en.wikipedia.org/wiki/Baire_space_%28set_theory%29) $\omega^\omega$. The latter meaning seems to be the one that makes sense here.2012-07-05
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    @Henning: Actually note that *all* those have the same cardinality. In the case of cardinal invariants, none of them actually matter and the question (at least in my eyes) is asking about the relation between $\frak c,d$ as cardinals.2012-07-05
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    @Asaf: Surely the cardinality is the same, but the notion of "dominates" you present here depends in which of them we're considering. For $2^\omega$, for example, there's a single function that dominates everything. It is not obvious to at least to me that what we have here is a property of the cardinality $\mathfrak c$ rather than a property of the specific additional structure of $\omega^\omega$.2012-07-05
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    @Henning: These cardinalities arise naturally from PCF theory, and there is a reason they are called cardinal invariants of the continuum.2012-07-05
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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.2012-07-05
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    @Brian: There are circling rumors (with very credible sources) that $\frak p=t$ was proved recently.2012-07-05
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    @Asaf: That would be interesting; do you know any more (e.g., by whom)?2012-07-05
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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.2012-07-05
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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.