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
8 events
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| Oct 12, 2021 at 3:39 | comment | added | Sridhar Ramesh | Saying "X is false in a nonstandard model of C, but is true in the standard model" is exactly the same as saying "X, but the system C I am interested in does not prove X". A person feels warranted to make this claim precisely when their metatheory T proves X, but also T proves "C does not prove X". | |
| Oct 12, 2021 at 3:38 | comment | added | Sridhar Ramesh | The language of nonstandard models perhaps ends up confusing this point. I think it would be easiest to understand the relevant dynamics focusing just on these brute syntactic things that you feel are clear. You should take "X holds in all models of logic system C" to mean exactly the same thing as saying "X is provable in logic system C". These might as well be synonymous phrases, for our purposes here. | |
| Oct 12, 2021 at 3:31 | comment | added | Sridhar Ramesh | If provability is clear to you, that's all that is going on. When people claim "X is true", all they are conveying is that they have a proof of X, in some theory they consider acceptable grounds for claiming things to be true (which is what people mean by "metatheory"). When people claim "X is true but unprovable (in system C)", what is going on is that their metatheory T proves X, but their metatheory T also proves "C does not prove X". Everything that's going on is, as you put it, "a finite amount of symbol manipulation". There's nothing else mystical going on. | |
| Oct 12, 2021 at 3:26 | comment | added | Sridhar Ramesh | "IF this logic system is consistent, THEN this logic system does not prove its Gödel sentence" is provable basically anywhere you like. It's provable in the logic system it refers to, and also it's provable in any other reasonable metatheory you care to consider. The proof is so basic that it goes through wherever you like. | |
| Oct 12, 2021 at 2:32 | comment | added | CouldntLoginToMyPreviousAcc | Provability has always been clear to me: it's just a finite amount of symbol manipulation. To me, a Gödel sentence (for PA) is just many symbols grouped in some syntactically correct way. The sentence has no meaning unless we assign meaning to its underlying symbols. So claims like "GS is true" sounded to me like "GS is true under all interpretations", hence my confusion. | |
| Oct 12, 2021 at 2:12 | comment | added | CouldntLoginToMyPreviousAcc | "IF this logic system is consistent, THEN this logic system does not prove its Gödel sentence" Is this sentence provable in the metatheory or in the logic system it refers to? | |
| Oct 12, 2021 at 0:33 | comment | added | Timothy Chow | Ha! Excellent. Somehow I missed that the OP started off by explicitly assuming consistency, like I missed the first sentence of the You are a bus driver puzzle. | |
| Oct 11, 2021 at 22:11 | history | answered | Sridhar Ramesh | CC BY-SA 4.0 |