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$\newcommand{\NS}{\mathrm{NS}}\mathcal{P}(\kappa)/I_{\NS}$ is the Boolean algebra of subsets of $\kappa$ under nonstationary symmetric difference, with $\mathbf{0} = [\emptyset] = I_{\NS}$, $\mathbf{1} = [\kappa] = I_{\NS}^*$ and $[X] \leq [Y]$ iff $X \setminus Y$ is nonstationary.

What is the maximum size of an ascending chain (well-ordered subset) in $\mathcal{P}(\kappa)/I_{\NS}$? Trivially it is at most $2^\kappa$, so $\kappa^+$ if $2^\kappa = \kappa^+$; is the $\kappa^+$ bound unconditional? Or can ascending chains of size $\kappa^{++}$ exist (from modest large cardinal strength)? This feels like something that would’ve been explored in the literature already but I can’t find any.

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    $\begingroup$ It is not obvious you can get an ascending chain of length $\kappa^+$. Whether this is possible is the question of the saturation of nonstationary ideal. $\endgroup$ Commented Sep 22 at 10:42
  • $\begingroup$ I was mostly thinking of large cardinals (e.g. weakly compact) $\kappa$, where an ascending chain of length $\kappa^+$ is easy. But, yeah, I suppose it’s interesting for $\kappa = \omega_1$ and such. $\endgroup$ Commented Sep 22 at 11:58
  • $\begingroup$ Anyway Gitik–Shelah showed ascending chains of length $\kappa^+$ exist (i.e., the nonstationary ideal is not saturated) for every regular uncountable $\kappa > \omega_1$. $\endgroup$ Commented Sep 22 at 13:06
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    $\begingroup$ Oh right, I just realized that my intuition of "ascending" / "descending" was muddled up. I had seen a characterization of saturation in terms of descending chains before and didn't realize that was the same as what I was asking for here. $\endgroup$ Commented Sep 22 at 16:29

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It is consistent. Assume GCH. First, by iterating the $\kappa$-closed analogue of Hechler forcing, add an ascending chain in $\kappa^\kappa / \mathrm{bd}$ of length $\kappa^{++}$, and the areas under the graphs of these functions form an ascending $\subseteq^*$-chain in $\kappa \times \kappa$. Mapping these to subsets of $\kappa$ we get an ascending $\subseteq^*$-chain $\langle A_\alpha : \alpha < \kappa^{++} \rangle$ of subsets of $\kappa$.

Now add one Cohen subset of $\kappa$, getting a $\diamondsuit_\kappa$-sequence $\langle d_\alpha : \alpha<\kappa \rangle$. For each $\beta<\kappa^{++}$, let $B_\beta \subseteq \kappa$ be $\{ \alpha<\kappa : d_\alpha \subseteq A_\beta \}$. By the properties of $\diamondsuit$, each $B_\beta$ is stationary. It is also strictly increasing mod $\mathrm{NS}$ because if $\beta<\gamma$, then $\{ \alpha : d_\alpha = (A_\gamma \setminus A_\beta) \cap \alpha \}$ is stationary.

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    $\begingroup$ Wait, how is $B_\beta$ related to $\beta$? Is $d_\alpha \subseteq A_\alpha$ a typo of $d_\alpha \subseteq A_\beta$? $\endgroup$ Commented Sep 23 at 7:44
  • $\begingroup$ @JaydeSM Oh right, typo. Thanks. $\endgroup$ Commented Sep 23 at 11:20

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