Timeline for Is it necessary that model of theory is a set?
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12 events
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| Mar 23, 2023 at 22:43 | comment | added | Joel David Hamkins | @MarcosCramer The internal models are not necessarily transitive, so there is no contradiction. | |
| Mar 23, 2023 at 22:00 | comment | added | Marcos Cramer | @JoelDavidHamkins In the last paragraph you claim that every model of ZFC contains a model of ZFC. As far as I can see, this can then be iterated indefinitely, leading to a violation of the foundation axiom. Where is the mistake in this reasoning? | |
| Nov 18, 2013 at 15:27 | comment | added | Joel David Hamkins | I misspelled Brice's name: it should be Brice Halimi, and his 2009 manuscript is entitled "Models and structures". | |
| Feb 18, 2010 at 16:38 | comment | added | kakaz | Thanks for that remark. Maybe there is possibility to extend such structure in other nonstandard-logic like paraconsistent one for example or higher order theories, which as far as I know gives some serious problems. However I learn a lot from Your answer and I wish to thank You a lot, because gives me a point of view which I may use in order to learn something from literature: it gives some kind of anchor or order in looking for other phenomena;-) | |
| Feb 18, 2010 at 16:23 | comment | added | Joel David Hamkins | Thanks for accepting my answer, although I sense that you are not fully satisfied, for which I am sorry. You had asked for something weaker than having a set model, but the completeness theorem shows that if there is no set model, then the theory is actually inconsistent. Thus, there is very little room for a weaker notion; you would inevitably be pushed to consider non-set "models" that in some sense satisfy an inconsistent theory... | |
| Feb 18, 2010 at 15:23 | comment | added | kakaz | So if I well understand, at least for first order theory completeness and "to be implemented as a set" is completely equivalent, assuming that by "set" we mean abstract implementation of ZFC axioms. | |
| Feb 18, 2010 at 15:20 | vote | accept | kakaz | ||
| Feb 18, 2010 at 15:07 | comment | added | kakaz | @Joel:"Perhaps you believe that if M is a model of ZFC, then it must be closed under all the set-building operations that exist in V" No I know that, model of ZFC in V may be not closed in V. But I have idea that if it is model inside some V and then is a set in the meaning that meets ZFC axioms, then it is also a set in the meaning of some other model of ZFC for which V is a "closed structure", then a set. So as such structure would exist it will lead to contradiction. I know that: "Thus, M's version of the power set of X may be much smaller than the power set of X as computed in V" Thank You | |
| Feb 18, 2010 at 15:04 | comment | added | François G. Dorais | If I could, I would add another +1 (or more) for the "final subtle point"! | |
| Feb 18, 2010 at 14:54 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Feb 18, 2010 at 14:36 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Feb 18, 2010 at 14:25 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |