As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency of measurable cardinals and $ZFC$:

As Prof. Robert Solovay recently put it: "For at least the last 20 years, Jack was convinced that measurable cardinals (and indeed $ZFC$) was inconsistent. He strove mightily to prove this. If he had succeeded it would have been the theorem of the century (at least) in set theory."

I was curious to find out what convinced him to not believe consistency of $ZFC$ and what I kind of attempts he tried, of course I found nothing.

Is there any published or unpublished note about his belief and approach, or possibly his philosophy toward it?

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    $\begingroup$ A tiny bit of anecdotal information may be found here: cs.nyu.edu/pipermail/fom/2007-August/011835.html $\endgroup$ – Timothy Chow Jan 30 '17 at 19:28
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    $\begingroup$ Silver had this idea that measurability implies the existence of cardinals with ("Ramsey-theoretic") properties that were too strong to be consistent. It seems his development of $0^\sharp$ was already an attempt to pursue and formalize this idea. He worked on it (privately) for many more years after he stopped publishing in set theory. Don't know of any writings (private or otherwise) where this is stated explicitly, though. $\endgroup$ – Andrés E. Caicedo Jan 30 '17 at 20:15
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    $\begingroup$ I believe Dominic McCarty talked to Silver less than 10 years ago about some of this. If you can get in touch with him, he may recall additional details. $\endgroup$ – Andrés E. Caicedo Jan 30 '17 at 21:05
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    $\begingroup$ If a notable set theorist labored on this approach for 20 years with so little to show...I'm surprised that there are so many votes to hear about it. $\endgroup$ – Matt F. Feb 1 '17 at 1:08
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    $\begingroup$ @ThomasBenjamin: You are somehow trying to blame the possible inconsistency of ZF on the Axiom of Infinity. ZF-Infinity being consistent and ZF being inconsistent does not tell you that the troublesome axiom is Infinity. It only says the axioms of ZF combined creates a problem. If you "believe" in the existence of $V_{\omega+\omega}$, then ZF-Replacement is consistent and hence the possible inconsistency of ZF comes from Replacement. $\endgroup$ – Burak Feb 2 '17 at 13:55

There is rumor that Silver's efforts started as an attempt to come up with a flawless argument showing the main "theorem" in Jensen's "A modest remark." That main "theorem" says that in ZF, there is no measurable cardinal. Magidor found the mistake a few days after Jensen released his manuscript. The manuscript still exists, but I'm pretty sure Jensen is happy if it's no longer circulated, even though it's historically also interesting as it has the first account of what's now called the Dodd-Jensen core model.

  • $\begingroup$ Fascinating! I don't want to go against Jensen's wishes, but would you be willing to describe the general idea behind his argument? $\endgroup$ – Noah Schweber Sep 4 '18 at 18:56
  • $\begingroup$ Very interesting historical remark! $\endgroup$ – Rahman. M Sep 4 '18 at 22:42

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