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I'm in search of a published proof that there is a (class) iteration that forces $\Diamond_\kappa(S)$ to hold for all regular uncountable $\kappa$ and all stationary $S \subseteq \kappa$.

It's OK if the citation assumes GCH. Surprisingly, I've been unable to find such a thing. Any suggestions?

Easton support iteration of adding Cohen sets at uncountable regular cardinals works, and the proof is routine to a set theorist. But the intended audience of the paper I'm writing may not know iterated forcing, and I'd prefer to just give a citation.

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  • $\begingroup$ Overkill, but can't you use Jensen coding to extend to a model L[x] for x a real? That's probably citable... ;) $\endgroup$
    – Todd Eisworth
    Commented Mar 6, 2021 at 18:33
  • $\begingroup$ @ToddEisworth ha ha, yes that does answer the question as posed, but I really need it to be an Easton support iteration (so that I can cite another black box about progressively closed Easton support iterations). $\endgroup$
    – Sean Cox
    Commented Mar 6, 2021 at 18:54
  • $\begingroup$ I guess I can also just put in into a short appendix, so people who don't want to think about forcing iterations can happily never encounter it... $\endgroup$
    – Sean Cox
    Commented Mar 6, 2021 at 18:55
  • $\begingroup$ Maybe more helpful: The construction of forcing GCH using class forcing is due to Jensen, in an abstract of the Notices that doesn't appear in Mathscinet. See Exercise 15.15 in Jech's 3rd Millenium edition and references at end of his chapter. If you can find a reference for why the collapses he uses force diamond, you're done. $\endgroup$
    – Todd Eisworth
    Commented Mar 6, 2021 at 18:59
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    $\begingroup$ Thanks. Diamond easily holds at successor cardinals after Jensen's forcing, but it's not obvious that it holds at inaccessibles. A math fairy sent me an email with some abstract reasons this should be true for those type of iterations, but I'd prefer a more explicit adding of diamonds everywhere (i.e. at all regulars). I think I'll just write a very brief appendix on it. $\endgroup$
    – Sean Cox
    Commented Mar 6, 2021 at 19:22

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