Timeline for Proof-theoretic comparison table?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2022 at 12:46 | comment | added | user21820 | @Trebor: I meant full LEM, which is not known to be inconsistent with MLTT and CIC. And I am not interested in HoTT in this post. Either way, I don't understand why you don't acknowledge that there is no double-negation translation for MLTT/CIC, which makes your initial comment wrong. | |
| Mar 30, 2022 at 10:16 | comment | added | Trebor♦ | Of course it depends on what you call LEM. If you interpret it naively, then HoTT + LEM = Falso, which is the most dramatic strength increase you can name. In practice we mean some version of truncated LEM, which is fine. | |
| Mar 30, 2022 at 10:05 | comment | added | user21820 | @Trebor: No, double-negation translation works for only certain systems, such as pure FOL to pure IFOL, or PA to HA. Even for FOL theories, it does not imply that adding LEM does not change the strength. Over the base theory IFOL we have ( A+LEM ⊢ Q ) iff ( DN(A) ⊢ DN(Q) ) where DN is the double-negation translation, which is not the same as ( A ⊢ DN(Q) ). In fact, CZF is quite weak compared to ZFC but CZF+LEM = ZFC. | |
| Mar 30, 2022 at 9:39 | comment | added | Pierre-Marie Pédrot | @Trebor there is no such thing as a double negation for CIC. If you do it on a proof relevant sort, it will break dependent elimination. If you do it on SProp, it works but SProp can only be used to prove dead branches so you don't increase you expressivity in Type. | |
| Mar 30, 2022 at 2:36 | history | became hot network question | |||
| Mar 30, 2022 at 1:54 | answer | added | Loïc Pujet | timeline score: 28 | |
| Mar 29, 2022 at 23:42 | comment | added | Trebor♦ | Adding LEM doesn't increase the strength. This is seen by the double negation translation. | |
| S Mar 29, 2022 at 18:35 | review | First questions | |||
| Mar 29, 2022 at 19:06 | |||||
| S Mar 29, 2022 at 18:35 | history | asked | user21820 | CC BY-SA 4.0 |