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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Is there any published or unpublished note about his belief and approach, or possibly his philosophy toward it?
Andrej Bauer
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Rahman. M
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  • Prof. Solovay is actually on MO. Perhaps he may be able to explain the motivation behind Silver's beliefs. – Burak Jan 30 '17 at 19:16
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    A tiny bit of anecdotal information may be found here: https://www.cs.nyu.edu/pipermail/fom/2007-August/011835.html – Timothy Chow Jan 30 '17 at 19:28
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    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. – Andrés E. Caicedo Jan 30 '17 at 20:15
  • @TimothyChow Thank you for the link. – Rahman. M Jan 30 '17 at 20:57
  • @AndrésE.Caicedo Today even we consider measurable cardinals as mid large cardinal, I don't understand strongness of Ramsey theoretic properties. Btw it's interesting. – Rahman. M Jan 30 '17 at 20:59
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    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. – Andrés E. Caicedo Jan 30 '17 at 21:05
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    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 01 '17 at 01:08
  • Since $ZF$ with the Axiom of Infinity dropped interprets $PA$, and $PA$ is consistent, the only way one could prove the inconsistency of $ZF$ (aside from showing that the standard proof of the consistency of $PA$ is wrong) is showing that adding the Axiom of Infinity to "$ZF$ with the Axiom of Infinity dropped" results in the inconsistency. Since "$ZF$ with the Axiom of Infinity dropped" should (like $PA$) prove the totality of its successor ($x$ $\cup$ {$x$}), adding Infinity seems (at first glance) tantamount to placing curly brackets around the finite ordinals and declaring that "whole" – Thomas Benjamin Feb 02 '17 at 07:36
  • (cont.) a "set". In the absence of any known paradoxes regarding the existence of an inductive set, where could a contradiction possibly come from? Did Prof. Silver believe that $PA$ was inconsistent? Is there any evidence to that effect? – Thomas Benjamin Feb 02 '17 at 07:41
  • (cont.) If not, did he believe that the assumption of the existence of an inductive set created a paradox? – Thomas Benjamin Feb 02 '17 at 07:49
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    @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. – Burak Feb 02 '17 at 13:55
  • @Burak:" Almost. Since "$ZF$ with the Axiom of Infinity dropped" should be able to prove "If the set $x$ exists then the set $x$$\cup${$x$} exists for all finite sets $x$" (the finite set $x$ exists because it can be constructed in a finite number of steps--apologies to Ultrafinitists), one already has an inductive 'collection' which already exists (the collection of all finite ordinals). The assumption the Axiom of Infinity makes is that this collection is a set (and not, say, a proper class). That the collection of all finite ordinals is a set seems, to me, more 'believable' – Thomas Benjamin Feb 02 '17 at 18:06
  • (cont.) than the inconsistency of $ZF$. This, I believe, is a basic intuition that leads many to believe in the consistency of $ZF$, and without a glimmer of a paradox on the horizon to challange that belief, why should that belief change? The same type of argument, I belive, would also hold for $V_{\omega + \omega}$ and Replacement. – Thomas Benjamin Feb 03 '17 at 03:41
  • As for Replacement, it is known that Replacement holds in $V_{\kappa}$ if $\kappa$ is inaccessible (see Samuel Coskey's senior thesis paper, "Partial Universes and the Axioms of Set Theory", Section 5.4, in particular, Theorem 5.15). In fact, Coskey states in Section 5.5 of his paper, "From the above discussion, it is clear that $V_{\alpha}$ is a model for all of $ZFC$ if $\alpha$ is an inaccessible cardinal which is larger than $\omega$. Since all of the arguments of the previous section are immune to the pass to the Axiom of Second Order Replacement, this statement can in fact be – Thomas Benjamin Feb 03 '17 at 08:24
  • (cont,) be strengthened to "if and only if" when this axiom is assumed." So it would seem that Replacement is not responsible for any inconsistency; rather, it would suggest that, in this case, $ZFC$ + "There exists an inaccessible cardinal" would be responsible for the inconsistency if an inconsistency were to exist. But again, where are the paradoxes (or even philosophical problems with the assumption of an inaccessible cardinal)? – Thomas Benjamin Feb 03 '17 at 08:41
  • @ThomasBenjamin: Re "and PA is consistent" in your comment above: would you please edit this to "and PA is only known to be consistent relative to ZF (or: relative to some considerably weaker systems)"? (You very likely know this, but some readers may get confused by the "absolute" claim in the current version of your comment. It is easy to improve comments, even if the 10-min-interval has passed: just copy, delete, and comment again.) – Peter Heinig Jul 16 '17 at 19:58
  • @PeterHeinig: what I actually should put is "$PA$ is only known to be consistent relative to the consistency of $PRA$ + $TI_{\epsilon}$ (as per Gentzen). What would be interesting to find out is, what fragment of $ZF(C)$ is equiconsistent with $PRA$ +$TI_{\epsilon}$ and in what sense (if any) can $PRA$ + $TI_{\epsilon}$ be deemed to "finitist' ? – Thomas Benjamin Jul 18 '17 at 15:09
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    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. – Tim Carlson May 02 '22 at 20:13

2 Answers2

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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.

Ralf Schindler
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    Fascinating! I don't want to go against Jensen's wishes, but would you be willing to describe the general idea behind his argument? – Noah Schweber Sep 04 '18 at 18:56
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    Very interesting historical remark! – Rahman. M Sep 04 '18 at 22:42
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    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," – Andreas Blass Apr 25 '21 at 00:21
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    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!" – Ralf Schindler Apr 25 '21 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.

Joel David Hamkins
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  • Brice tells me that earlier versions of his theorem are due to Suzuki & Wilmers and also John Schlipf. – Joel David Hamkins Apr 11 '23 at 14:12
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    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. – Peter LeFanu Lumsdaine Apr 11 '23 at 16:09
  • @PeterLeFanuLumsdaine Unfortunately, I can't speak for Silver, and as I mentioned, this is all second-hand, and so I am uncertain of his actual views. I agree with you that this mistake is most commonly seen in erroneous applications, rather than in the theory itself. Meanwhile, I do find the issue quite subtle, and it doesn't strike me as unreasonable to think that since we often make this mistake, perhaps there is a subtle logical paradox to be found here. For example, the case of measurable cardinals involves elementary embeddings, which bring the two semantical perspectives into conflict. – Joel David Hamkins Apr 11 '23 at 16:15
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    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. – James Hanson Apr 12 '23 at 06:17
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    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". – Ali Enayat Apr 29 '23 at 21:06
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    Thanks, Ali. Meanwhile, Brice Halimi's paper is available at https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-58/issue-1/Models-as-Universes/10.1215/00294527-3716058.full. 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. – Joel David Hamkins Apr 29 '23 at 23:13
  • Thanks for the references Joel. I remember reading the Suzuki-Wilmers paper as a graduate student, after all these years, it is still worth reading (but very hard to get hold of). – Ali Enayat Apr 30 '23 at 01:51