# Silver's approach to the inconsistency of $\mathrm{ZFC}$

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 $$\mathrm{ZFC}$$:

As Prof. Robert Solovay recently put it: "For at least the last 20 years, Jack was convinced that measurable cardinals (and indeed $$\mathrm{ZFC}$$) 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 $$\mathrm{ZFC}$$ 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##### Is there any published or unpublished note about his belief and approach, or possibly his philosophy toward it?
• A tiny bit of anecdotal information may be found here: cs.nyu.edu/pipermail/fom/2007-August/011835.html Jan 30, 2017 at 19:28
• 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. Jan 30, 2017 at 20:15
• 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. Feb 1, 2017 at 1:08
• @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. Feb 2, 2017 at 13:55
• I was a postdoc at Berkeley for 1980-82. Silver gave a talk on his general approach to proving inconsistency in the logic colloquium at some point during that time. Unfortunately, I remember very little of the talk except that it was mostly about an approach rather than explicit results. It was well attended, so maybe some notes will turn up someday. May 2 at 20:13