Timeline for Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?
Current License: CC BY-SA 4.0
9 events
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| May 26, 2022 at 14:20 | comment | added | Emil Jeřábek | @AlexKruckman A theory $T$ is reflexive if it proves the consistency of each finite subtheory of $T$. And $T$ is (locally) essentially reflexive if all its extensions (wlog finite) are reflexive, or in other words, if it proves the reflection schema $\Box_{T_0}(\phi)\to\phi$ for all finite subtheories $T_0\subseteq T$ and sentences $\phi$. A useful sufficient condition is that all sequential theories that prove full induction are (even uniformly) essentially reflexive. | |
| May 26, 2022 at 13:56 | comment | added | Alex Kruckman | What is a reflexive theory? | |
| May 26, 2022 at 11:19 | comment | added | Emil Jeřábek | You are welcome. | |
| May 26, 2022 at 11:06 | comment | added | Joel David Hamkins | Thank you very much for this extremely informative answer! Perfect. | |
| May 26, 2022 at 10:59 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
I believe Lindström misreports this. The argument needs essential reflexivity. For example, $PRA + \neg Con(I\Sigma_1)$ is reflexive and $\Sigma_1$-unsound, yet it is trustworthy by Visser’s results.
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| May 26, 2022 at 10:10 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| May 26, 2022 at 10:03 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| May 26, 2022 at 9:49 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
I messed up the references
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| May 26, 2022 at 7:16 | history | answered | Emil Jeřábek | CC BY-SA 4.0 |