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Sep 1, 2021 at 7:38 comment added Jason Zesheng Chen With respect to the specific problem of requiring a specific closure, this is treated for instance in Jech by Lemma 21.9.
Sep 1, 2021 at 7:36 comment added Jason Zesheng Chen Since we'll need to disturb the continuum function very often below $\kappa$, and we won't know beforehand what the last bit of the forcing will look like (i.e. the poset named by the final factor in the iteration, to add $\kappa^{++}$ subsets to $\kappa$), so we let iteration take care of that by letting the final factor just be "that poset to add this many subsets to $\kappa$ when we get there".
Sep 1, 2021 at 7:19 history edited Jason Zesheng Chen CC BY-SA 4.0
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Sep 1, 2021 at 7:19 comment added Reflecting_Ordinal Thank you. And how does the iterated forcing solved this problem?
Sep 1, 2021 at 7:16 history answered Jason Zesheng Chen CC BY-SA 4.0