Timeline for Silver's approach to the inconsistency of $\mathrm{ZFC}$
Current License: CC BY-SA 4.0
12 events
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| Apr 30, 2023 at 1:51 | comment | added | Ali Enayat | 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). | |
| Apr 29, 2023 at 23:13 | comment | added | Joel David Hamkins | 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. | |
| Apr 29, 2023 at 21:06 | comment | added | Ali Enayat | 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". | |
| Apr 21, 2023 at 12:19 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 12, 2023 at 6:17 | comment | added | James E Hanson | 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. | |
| Apr 11, 2023 at 21:13 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 11, 2023 at 16:15 | comment | added | Joel David Hamkins | @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. | |
| Apr 11, 2023 at 16:09 | comment | added | Peter LeFanu Lumsdaine | 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. | |
| Apr 11, 2023 at 14:12 | comment | added | Joel David Hamkins | Brice tells me that earlier versions of his theorem are due to Suzuki & Wilmers and also John Schlipf. | |
| Apr 11, 2023 at 13:11 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 11, 2023 at 13:01 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 11, 2023 at 12:47 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |