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Is there a set $S$ of $\mathbb \aleph_1$ sequences of natural numbers such that for any sequence not in $S$, there is a sequence in $S$ that grows faster than it?

Assuming the continuum hypothesis this is trivially true (just take $S$ to be the set of all sequences), so we'll work on the assumption that the continuum hypothesis is false.

My first thought is that $S$ would need to be similar to a fast-growing hierarchy, but ranging over all the countable ordinals instead of just some of them.

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    $\begingroup$ There is a whole subject about this kind of question, called cardinal characteristics of the continuum. What you have is dominating family, and the size of the smallest dominating family is called the dominating number. It can be strictly larger larger than $\omega_1$. $\endgroup$
    – Joel David Hamkins
    Commented Oct 29, 2023 at 21:12
  • $\begingroup$ In case you do not already know about this, you should read about cardinal characteristics of the continuum: depending exactly on what you mean by “grows faster”, you seem to be asking whether the dominating number $\mathfrak{d}$ equals $\aleph_1$, something which is undecidable in ZFC. $\endgroup$
    – Gro-Tsen
    Commented Oct 29, 2023 at 21:13
  • $\begingroup$ See mathoverflow.net/a/29659/1946, for example, or mathoverflow.net/a/9027/1946, or many other MO questions. $\endgroup$
    – Joel David Hamkins
    Commented Oct 29, 2023 at 21:13

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