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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Is there any published or unpublished note about his belief and approach, or possibly his philosophy toward it?
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    $\begingroup$ A tiny bit of anecdotal information may be found here: cs.nyu.edu/pipermail/fom/2007-August/011835.html $\endgroup$ Jan 30, 2017 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. $\endgroup$ Jan 30, 2017 at 20:15
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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. $\endgroup$
    – user44143
    Feb 1, 2017 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, 2017 at 13:55
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    $\begingroup$ 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. $\endgroup$ May 2, 2022 at 20:13

2 Answers 2

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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.

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    $\begingroup$ Fascinating! I don't want to go against Jensen's wishes, but would you be willing to describe the general idea behind his argument? $\endgroup$ Sep 4, 2018 at 18:56
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    $\begingroup$ Very interesting historical remark! $\endgroup$
    – Rahman. M
    Sep 4, 2018 at 22:42
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    $\begingroup$ Some years ago, waiting at a train station after a set theory conference, I overheard Solovay asking Jensen whether he still had any copies of "A Modest Remark." Jensen's answer was "I certainly hope not," $\endgroup$ Apr 25, 2021 at 0:21
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    $\begingroup$ I know that Jensen does have a copy, at least on DVD, as Steel and me gave him a DVD with all of his papers and handwritten notes on the occasion of his 70th birthday. The modest remark paper was one of them, and when he [Jensen] realized, he said: "Oh, that's my favorite paper!" 😁 $\endgroup$ Apr 25, 2021 at 11:59
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When I was a graduate student at Berkeley in the early 90s, I had heard that Silver's approach to refuting ZFC involved the idea that somehow we make a mistake in our thinking about ZFC by conflating the internal idea of syntax and semantics, as undertaken internally to the object theory by representing formulas and syntax trees etc. as sets and then defining the notion of truth for them as in the Tarski recursion, and the external metatheoretic notions of syntax and semantics. Silver's idea, as I understood it, was that there was some subtle logical error that arises in our thinking about this.

This is all second-hand, and I never heard this view directly from Silver, however, as he rarely talked about his project.

The two notions do definitely pull apart in $\omega$-nonstandard models, and there are many interesting observations to make about the difference. For example, there is the beautiful theorem of Brice Halimi that every model of ZFC has an element that is a structure $\langle M,\varepsilon\rangle$, which when viewed externally as a model in the language of set theory, satisfies ZFC. In short, every model of ZFC has an element that is a model of ZFC. Understanding the statement and proof of this theorem requires one to distinguish carefully between the object-theoretic ZFC and the meta-theoretic ZFC, since the original model, of course, might satisfy ¬Con(ZFC), and yet still it has a model $\langle M,\varepsilon\rangle$ that satisfies every actual axiom of ZFC.

Meanwhile, let me also state as a fact that many mathematicians and set theorists do commonly conflate the internal and external approaches. Let me give three examples.

Truth predicates. One can commonly find incorrect definitions of what it means to have a class truth predicate. Namely, people sometimes say that a truth predicate in set theory is a class $T$ of pairs $\langle\varphi,\vec a\rangle$ such that $V\models\varphi[\vec a]$ or something like that, which is trying to use the external semantic notions internally. To formalize truth in Gödel-Bernays set theory or Kelley-Morse, however, one should speak of a satisfaction class that fulfills the disquotational Tarski recursion, and this will involve nonstandard formulas.

Forcing relation. Similar confusions surround much of the literature on the definition of the forcing relation, particularly in the case of class forcing. In many accounts, people try to define the forcing relation by saying $p\Vdash_{\mathbb{P}}\varphi$ over $M$ if for every $M$-generic filter $G\subset\mathbb{P}$ we have $M[G]\models\varphi$. But this account, which uses the meta-theoretic notion, is not generally correct; for example, some uncountable models $M$ may have no generic filters. Certainly it is not correct for defining the forcing relation over $V$ in a model of Gödel-Bernays set theory. The confusion often causes people to think incorrectly that we have a satisfactory forcing technology only over countable transitive models.

Meanwhile, in my paper with several co-authors

we provide what I view as the correct definition of what it means to have a forcing relation, namely, it means to have a class relation that fulfills the forcing recursion. This is exactly analogous to the difference between the internal/external account of what it means to have a truth predicate.

Elementarity of embeddings. In large cardinal set theory one often wants to refer to elementary embeddings defined on proper class domains or often $V$ itself, $j:V\to M$. But how does one express the elementarity requirement? Typically, in the literature one sees people say that $j$ is $\Sigma_1$-elementary and cofinal. This implies by induction (in the metatheory) that $j$ preserves the truth of any $\Sigma_n$ assertion, for any metatheoretically finite $n$. So one has that the embeddings are elementary, but only for meta-theoretically finite assertions, and so this notion falls short of what might be regarded as fully elementary. It is indeed a subtle matter to get elementarity for all assertions in the object theory. So again we see the distinction between the internal syntax/semantics and the external.

My point with all these examples is that people do sometimes make the mistake that I have claimed Silver highlights in his proof strategy.

Nevertheless, what is much less clear to me is how one will turn this into a refutation of ZFC, since I don't see the error being made in the axiom schemes of ZFC itself, but rather only in mistaken applications of it.

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  • $\begingroup$ Brice tells me that earlier versions of his theorem are due to Suzuki & Wilmers and also John Schlipf. $\endgroup$ Apr 11, 2023 at 14:12
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    $\begingroup$ Can you elaborate at all (or point to any elaboration) on how Silver saw this subtlety as a possible strategy for proving inconsistency? I’m much more used to seeing subtleties like this play the opposite rôle — the mistaken conflation of external/internal syntax gives an apparent inconsistency (as e.g. this recent question), and the careful distinction between these notions shows up the gap in the claimed inconsistency proof. $\endgroup$ Apr 11, 2023 at 16:09
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    $\begingroup$ I find it interesting that this is similar to Edward Nelson's concerns about the consistency of PA. My understanding is that he thought there was some subtlety with the validity of arithmetizing proof systems. $\endgroup$ Apr 12, 2023 at 6:17
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    $\begingroup$ Sorry for this tardiness of this comment: another source for the result attributed to Halimi is the 1974 paper of Claes Åberg (published in Synthese) entitled "Relativity phenomena in set theory". $\endgroup$
    – Ali Enayat
    Apr 29, 2023 at 21:06
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    $\begingroup$ Thanks, Ali. Meanwhile, Brice Halimi's paper is available at projecteuclid.org/journals/notre-dame-journal-of-formal-logic/…. The Suzuki/Wilmers result he credits (which seems to predate Åberg) is Suzuki, Y., and G. Wilmers, “Non-standard models for set theory,” pp. 278–314 in The Proceedings of the Bertrand Russell Memorial Logic Conference (Uldum, 1971), edited by J. L. Bell, J. C. Cole, G. Priest, and A. B. Slomson, Bertrand Russell Memorial Logic Conference, University of Leeds, Leeds, 1973. MR 0351814. $\endgroup$ Apr 29, 2023 at 23:13

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