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In Friedman and Magidor - The Number of Normal Measures, the authors use a nonstationary support iteration of posets, rather than the more customary countable or Easton support iterations: conditions $p$ are allowed where $\operatorname{supp}(p) \cap \lambda$ is nonstationary in $\lambda$ for all inaccessible $\lambda \leq \kappa$. Thus, all conditions which are allowed in an ESI are allowed here, and then some.

I unfortunately haven't had the opportunity to inspect the details of this paper. Does anybody know of other applications of nonstationary support iterations and what makes them substantially different from ESIs?

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    $\begingroup$ Nonstationary support iterations turn out to be useful because they allow fusion arguments to work. In other words, while a nonstationary support iteration of length $\kappa$ will probably not have substantial closure (just like an Easton support iteration), you can still prove that carefully constructed length $\kappa$ sequences have lower bounds. $\endgroup$ Commented Nov 13 at 0:04
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    $\begingroup$ Also, see this recent paper by Kaplan where he simplifies the Friedman--Magidor argument to just use nonstationary support products, instead of iterations. $\endgroup$ Commented Nov 13 at 0:05
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    $\begingroup$ @MihaHabič , why not post an answer? $\endgroup$ Commented Nov 13 at 1:28
  • $\begingroup$ I see, thanks. I’ll accept it as an answer. $\endgroup$ Commented Nov 13 at 10:14

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Nonstationary-support iterations are useful because they allow for fusion arguments. If you have an iteration of length $\kappa$, you can rarely expect it to have any significant closure. A nice feature of Easton-support iterations is that they let you factor your iteration into a small part and a tail with ok closure. Nonstationary-support iterations are similar, but also let you get away with finding lower bounds of sequences of conditions of length $\kappa$ (provided that you build the sequence carefully).

I believe the first use of nonstationary-support iterations was Jensen's "coding the universe" theorem (probably not the best first example to look at, though). There are other, more recent papers using iterations like this, for example Gitik, Kaplan -- Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model.

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