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Looking over the various cardinal characteristics of the continuum all of them are defined by a sentence of the form:

"$\mathfrak{x}$ is the least cardinality of a subset of $\omega^\omega$ that $\ldots$"

Is there any meaningful cardinal characteristic whose definition can be given by

"$\mathfrak{x}$ is the supremum of the cardinalities of subsets of $\omega^\omega$ that $\ldots$"

I tried negating the definition of the usual cardinal characteristics, e.g. $\mathfrak{b},\mathfrak{d}$, but there are easy examples of size $2^{\aleph_0}$.

So, I am afraid the answer is "No", but I would be glad to hear otherwise.

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    $\begingroup$ What about the supremum of the cardinalities of strong-measure-zero subsets of $\mathbb R$? I don't know if you'd consider it a "cardinal characteristic" or not, especially since it is consistent that this cardinal is countable, but otherwise it fits the bill. There's a question about it here: mathoverflow.net/questions/281024/the-strong-measure-number/…. $\endgroup$
    – Will Brian
    Commented Jan 22, 2020 at 14:17
  • $\begingroup$ @WillBrian: I have to see what is the strong-measure-zero subsets of R. $\endgroup$
    – Ioannis Souldatos
    Commented Jan 22, 2020 at 18:38
  • $\begingroup$ @WillBrian. This seems a good candidate, but is it consistent that $\mathfrak{s}_+$ is a cardinal between $\aleph_0$ and $2^{\aleph_0}$? $\endgroup$
    – Ioannis Souldatos
    Commented Jan 22, 2020 at 18:55
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    $\begingroup$ It is a theorem that $\mathfrak{s}_+$ is a cardinal between $\aleph_0$ and $2^{\aleph_0}$ (inclusive). It is consistent that it is $\aleph_0$ (this happens in Laver's model -- google "Borel conjecture" to read more about it), that it is uncountable but less than $2^{\aleph_0}$ (this happens in the random real model), and that it is equal to $2^{\aleph_0}$, even when CH fails (this happens in the Cohen model). $\endgroup$
    – Will Brian
    Commented Jan 22, 2020 at 19:06
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    $\begingroup$ The results about the Cohen and random model can be found in chapter 8 of Bartoszynski and Judah's book Set Theory: on the structure of the real line. For the random real model, it follows from Theorem 8.2.8 that there are uncountable strong measure zero sets, and it is Theorem 8.2.11 that every strong measure zero set has size $\leq \aleph_1$. $\endgroup$
    – Will Brian
    Commented Jan 23, 2020 at 13:52

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