Timeline for Why do people say Gödel's sentence is true when it is true in some models but false in others?
Current License: CC BY-SA 4.0
22 events
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| Oct 11, 2021 at 15:13 | comment | added | Mike Shulman | Thanks for the clarification. I agree with what you wrote. My point was the same as what you said: when T is as strong or stronger as "the theory of informal mathematics", it is not justified to simply assert "The Godel sentence for T is true" without qualification. | |
| Oct 11, 2021 at 15:12 | comment | added | Timothy Chow | @OscarSmith The axiom of choice is not needed. In fact, almost nothing is needed to prove Gödel's incompleteness theorem. | |
| Oct 11, 2021 at 4:29 | comment | added | Oscar Smith | Does godel's sentence rely on the axiom of choice? In other words, is ZF enough for it? | |
| Oct 10, 2021 at 20:27 | comment | added | Andrej Bauer | @PeterLeFanuLumsdaine: yes, I mean the meta-theory (hence my saying "the one your brain is imagining"), although it takes a bit of introspection to call it meta-theory. The bit about constructing a model $O$ of the object-theory in which Gödel's sentence $g$ fails, it is helpful to note that $O$ depends on $g$. | |
| Oct 10, 2021 at 19:25 | comment | added | Peter LeFanu Lumsdaine | @AndrejBauer: To forestall misunderstanding — When you say “any model”, do you mean “…of your meta-theory/foundations”, or “…of the object theory”? I guess you mean the former, in which case I agree entirely with your comment. But I think in speaking of the “standard model”, OP and others here mean “…of the object theory” or “…of arithmetic”, and for that, Gödel’s paper certainly doesn’t show that the Gödel sentence is true in any model — it shows, of course, precisely the contrary. | |
| Oct 10, 2021 at 18:19 | comment | added | Timothy Chow | @CouldntLoginToMyPreviousAcc The Gödel sentence for ZFC is equivalent to "ZFC is consistent." Anyone who believes that ZFC is a plausible foundation for mathematics a fortiori believes that ZFC is consistent. Conversely, skeptics about infinite set theory typically don't believe that ZFC is consistent. As far as proving ZFC is consistent is concerned, we can't do this on the basis of ZFC itself. Whether this means that the consistency of ZFC is "truly undecidable" is a philosophical question. | |
| Oct 10, 2021 at 16:37 | comment | added | Andrej Bauer | I would just like to point out that it does not matter at all what the "standard model" is, it may even be different for each person. Let $M$ be whatever model you wish to carry out the argument in, for instance the one your brain imagines to be imagining. Read Gödel's paper as if it is all taking place in $M$. At the end of it, the paper will have convinced you that a sentence $g$ has been constructed which is true in $M$. The same goes for any other statement, there is nothing special here about Gödel's theorem. | |
| Oct 10, 2021 at 15:35 | comment | added | Peter LeFanu Lumsdaine | @MikeShulman: I’ve added a note about how this applies to the Gödel sentence for ZFC. I don’t see any essential difference in that case; if you think something’s different, could you elaborate on what? | |
| Oct 10, 2021 at 15:33 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
Added note on Gödel sentence for ZFC and other theories
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| Oct 10, 2021 at 13:25 | comment | added | CouldntLoginToMyPreviousAcc | One more question: what's the status of Gödel sentence for ZFC? Is it believed to be also true (but unprovable) or truly undecidable? | |
| Oct 10, 2021 at 8:29 | comment | added | user44143 | @WillSawin, there is also Andreas Blass’s argument for accepting ZFC as an appropriate axiom system for current mathematics: mathoverflow.net/a/90945/44143 | |
| Oct 9, 2021 at 23:18 | comment | added | Mike Shulman | As Alex says, this answer makes sense if by "the Godel sentence" one means the Godel sentence for PA. But the OP didn't specify that. It's not at all clear to me that one can apply this same argument to the Godel sentence for ZFC. | |
| Oct 9, 2021 at 22:40 | comment | added | Christopher King | A small correction: ZFC plus the hypotheses of Godel's incompleteness theorem imply the Godel statement. This is important because one of the hypotheses is that the theory is consistent, which ZFC might not be able to prove. | |
| Oct 9, 2021 at 16:56 | comment | added | Sam Hopkins | @CouldntLoginToMyPreviousAcc: See "Believing the axioms" by Maddy (cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf) | |
| Oct 9, 2021 at 14:59 | comment | added | Will Sawin | @CouldntLoginToMyPreviousAcc (1) The acceptance of ZFC rests on multiple pillars, including the fact that most theorems proven in ZFC match our intuitions, as well as the fact that the axioms of ZFC match our intuitions, and that many other reasonable axioms systems are closely related to ZFC. (2) It's possible that a better axiom system could be found although (in a take I read somewhere on this website, maybe from JDH) it's very unlikely that mathematicians would ever accept a resolution of CH since we know so much interesting math conditional on CH or its negation. | |
| Oct 9, 2021 at 14:52 | comment | added | CouldntLoginToMyPreviousAcc | Regarding ZFC being the standard, how do mathematicians know ZFC is the right axiomatic system to use? Is it because most of the theorems provable from ZFC match our mathematical intuitions/expectations? Regarding unprovable statements of ZFC such as CH, if someone devices a novel axiomatic system which proves or disproves CH and proves all the other interesting theorems such as FLT, would it be a better system and can therefore replace ZFC? | |
| Oct 9, 2021 at 14:42 | comment | added | CouldntLoginToMyPreviousAcc | Just to check if I understand correctly: Godel sentence is provable in ZFC and therefore true from ZFC, which is a more powerful and standard axiomatic system of mathematics. Since ZFC is the standard, we simply say Godel sentence is true. | |
| Oct 9, 2021 at 12:19 | comment | added | Alex Kruckman | +1. It would be a good idea to also be more precise about what you mean by "the Gödel sentence". There's a difference between the relationship between ZFC and the Gödel sentence for PA (for example) vs the Gödel sentence for ZFC itself. This could be related to the OP's confusion: are we justified in saying the Gödel sentence for ZFC is true? | |
| Oct 9, 2021 at 11:56 | comment | added | Peter LeFanu Lumsdaine | @AlessandroDellaCorte: Indeed, thanks for the catch! | |
| Oct 9, 2021 at 11:55 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
corrected typo, thanks Alessandro Della Corte in comments!
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| Oct 9, 2021 at 11:07 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
a little improvement
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| Oct 9, 2021 at 11:00 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |