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This is a historical question: Who was the first person to notice the following?

If $V \models \kappa$ is measurable and $P$ adds $\kappa$ Cohen reals, then in $V^P$, letting $\hat{I}$ to be the ideal generated by a normal prime ideal $I$ over $\kappa$, forcing with $\hat{I}$ is equivalent to adding $|2^{\kappa}|^V$ Cohen reals.

I always thought this was due to Prikry but I cannot find this in his thesis.

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  • $\begingroup$ Just curious, where did you first learn about this result? A book of Fremlin maybe? $\endgroup$
    – Monroe Eskew
    May 5, 2015 at 14:17
  • $\begingroup$ I learned this fact from Kunen. He thinks this may have been known in the 1960s but doesn't remember seeing this in print. Fremlin's book does have this theorem without any attribution (Measure theory, vol. 5 part II, 555G). I also asked Kanamori, Prikry and Solovay but they didn't recall any reference either. $\endgroup$
    – Ashutosh
    May 5, 2015 at 16:09
  • $\begingroup$ As you know this fact follows easily from Foreman's Duality Theorem. My hunch is that Foreman came up with a unification of various folklore arguments about induced ideals that had been floating around since forever. $\endgroup$
    – Monroe Eskew
    May 5, 2015 at 16:18
  • $\begingroup$ What about Solovay's argument for RVM cardinals? Does his proof easily translate to show the claim about adding measurable-many Cohen reals? $\endgroup$
    – Monroe Eskew
    May 5, 2015 at 16:24
  • $\begingroup$ Yes, this is how Kunen proved it. There is a sketch (Lemma 3.2) here: math.wisc.edu/~akumar/IND_IDEALS.pdf $\endgroup$
    – Ashutosh
    May 5, 2015 at 16:34

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