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Let $\mathsf{R}$ denote some finitely many relations about finitely many cardinal characteristics (e.g. $\mathfrak{a} \leq \mathfrak{s}$, $\mathfrak{a} < \mathfrak{d} = \mathfrak{r}$, $\mathfrak{b} = \mathfrak{s} \wedge \mathrm{cov}(\mathcal{M}) < \mathrm{non}(\mathcal{N})$ etc). Is there some such statement $\mathsf{R}$ such that

  • $\mathsf{ZFC} + \mathsf{R}$ is consistent, but
  • $\mathsf{ZFC} + \mathsf{R} + \mathfrak{c} < \aleph_\omega$ is inconsistent (or, at the very least, not known to be consistent)?
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  • $\begingroup$ $\mathfrak d < \mathfrak a$, maybe? I know that Shelah's original proof of this necessarily has $\mathfrak d >> \aleph_\omega$, and this was the state of the art for quite a while. I can't remember at the moment whether anyone has gotten $\mathfrak d < \mathfrak a < \aleph_\omega$ in the years since. I know it is still open whether $\aleph_1 = \mathfrak d < \mathfrak a$ is consistent. $\endgroup$
    – Will Brian
    Commented 55 mins ago
  • $\begingroup$ Nevermind. I searched around a bit, and it looks like Shelah used a template forcing to get $\aleph_2 = \mathfrak d < \mathfrak a = \aleph_3$. $\endgroup$
    – Will Brian
    Commented 45 mins ago

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