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Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.

At most papers Silver's result is stated as follows:

(1) If $\kappa$ is a $\kappa^{++}-$supercompact cardinal, then in a generic extension, $\kappa$ remains measurable and $2^\kappa = \kappa^{++}.$

But for example in the paper Strong axioms of infinity and elementary embeddings, Silver's result is stated as follows:

(2) If $\kappa$ is $\eta+\delta+1-$extendible, then there is a forcing extension, in which $2^\kappa=\aleph_{\kappa+\delta}$ and $\kappa$ is still $\eta-$extendible if $\eta>0$ and measurable if $\eta=0.$

Question 1. Which formulation, is Silver's original result?

I also wonder to know if there is any way of obtaining Silver's unpublished paper about reverse Easton iterations.

Question 2. Is the method of reverse Easton iteration also due to Silver?

I have read at some papers (I don't remember which papers) that the formulation of reverse Easton iteration was done also by some other people like Ronald Jensen and Franklin Tall (in his PhD thesis), but it was Silver who used the method in connection with large cardinals. If so, does anyone know where Jensen used this method in his work (of course for the first time)?

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    $\begingroup$ I never understood the reason for it to be called "reverse". $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 2, 2014 at 6:11
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    $\begingroup$ Maybe one reason is the following: Easton forcing can be imagined as an iteration from upward to downward (for example the product of $Add(\omega, \omega_2)\times Add(\omega_1, \omega_3)$ can be imagined as a two step iteration first with $Add(\omega_1, \omega_3)$ followed by $Add(\omega, \omega_2)$ (and you can not reverse the iteration). But in reverse Easton iteration, the iteration is from downward to upward. So you are reversing the kind of iteration!!!!! $\endgroup$
    – Mohammad Golshani
    Commented Sep 2, 2014 at 6:20
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    $\begingroup$ My opinion is that it should not be called "reverse", and we should all abandon that old terminology. The proper distinction is between an Easton-support product and an Easton-support iteration; in both cases we use support in the Easton ideal. Moreover, the Easton-support iteration is more forward than it is reversed, since you are taking the posets from the later stages as the iteration progresses. (This would be "reverse" only if one imagines walking through the iteration while facing backwards.) Please call this the Easton-support iteration, which accurately describes the ideal. $\endgroup$
    – Joel David Hamkins
    Commented Sep 2, 2014 at 11:10
  • $\begingroup$ @JoelDavidHamkins Thanks for your comment. Do you know who first used the name "reverse Easton iteration" for such kind of forcing iterations. Something maybe related to your note is the use of lemma for Jensen's covering theorem (Jensen's covering lemma); while it is a very deep theorem (and in fact Kanamori says it the most important theorem of set theory in 1970th). $\endgroup$
    – Mohammad Golshani
    Commented Sep 2, 2014 at 11:27
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    $\begingroup$ The "reverse" terminology goes way back -- I'm not sure who started it -- but it is increasingly on the way out. I made an argument against it at the Berkeley Logic Colloquium in the mid-1990s, and I remember walking backwards across the stage and then forwards to illustrate the point, with Silver, Woodin, Solovay and others in the audience. $\endgroup$
    – Joel David Hamkins
    Commented Sep 2, 2014 at 11:44

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