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In his paper "Forcing in admissible sets", Ershov writes

In unpublished lectures given at Novosibirsk State University in 1976-1977 on the theory of admissible sets, the author showed that it is possible to use forcing to construct admissible sets. The possibility of using forcing in the theory of admissible sets was suggested in a number of publications, but a precise exposition of this method was never published. The present paper, which is based on notes from the lectures mentioned above, attempts to fill this gap.

I'm curious about the extent of the "suggestions" that Ershov mentions. Specifically: would it be fair to credit Ershov solely with the result that set forcing preserves admissibility, or should that be Ershov/folklore (or something else)?

Context: I'm writing a paper where I use this result, and would like to attribute it appropriately.

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    $\begingroup$ I'm also very interested in this history. $\endgroup$
    – Joel David Hamkins
    Dec 22, 2016 at 0:22
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    $\begingroup$ I always assumed Barwise was involved. $\endgroup$
    – Joel David Hamkins
    Dec 22, 2016 at 1:05
  • $\begingroup$ @JoelDavidHamkins I would too. But in his book ASAS (published about a year before Ershov's lectures), forcing only appears twice, once in the introduction (where he mentions he's omitting treatment of it), and once at the very end of chapter IV, where he mentions that a couple results can be proved via forcing. Unless I'm missing something, he doesn't give any bibliographic references, and it looks like he personally hadn't published anything on the subject by the time of his book. $\endgroup$
    – Noah Schweber
    Dec 22, 2016 at 1:17

2 Answers 2

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Forcing over admissible sets was first carried out by Jon Barwise in his 1967 Stanford dissertation.

This 2015 paper of Mathias (it appeared in Fundamenta Mathamaticae) develops forcing over set theories that are weaker than KP. It also has a nice historical section (see 11.8, p.46) on the topic of forcing over weak set theories.

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  • $\begingroup$ Thanks! I've accepted since this seems the earliest source (although right around the time of Jensen's lectures). Do you happen to have a copy of Barwise's dissertation, or know where I could track one down? I've been having trouble finding it, since he shortly after published a paper with the same title but which only covered (IIUC) the first half or so of his thesis. $\endgroup$
    – Noah Schweber
    Jan 2, 2017 at 2:57
  • $\begingroup$ @NoahSchweber. I don't have a copy of Barwise's thesis, but I read a brief description of his forcing argument in the context of admissible sets in the paper by Zarach listed as [Z2] in the references of Mathias' paper in my answer; there Zarach refers to an appendix to Barwise's thesis were the forcing argument appeared. By the way, you might be able to get hold of the thesis through proquest.com $\endgroup$
    – Ali Enayat
    Jan 8, 2017 at 3:44
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Some of the early results in this erea are due to Jensen. See his notes "Admissible sets", available at his homepage "https://www.mathematik.hu-berlin.de/~raesch/org/jensen.html".

As the introduction indicates, they are based on lectures given at 1969 and section 5 of the paper deals with forcing over admissible sets (and even primitive recursive closed sets).

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    $\begingroup$ I've been reading those notes (specifically for their coverage of class forcing)! I misread the date, though; I thought they were from 1979. Thanks! $\endgroup$
    – Noah Schweber
    Dec 22, 2016 at 2:29

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