Timeline for Is the union of two conservative extensions of a theory conservative?
Current License: CC BY-SA 4.0
6 events
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| Apr 14, 2023 at 16:52 | comment | added | Holo | @EmilJeřábek good catch, I'll admit I'm usually loose with my use of LEM, so I tend to not notice "useless contradictions" | |
| Apr 14, 2023 at 16:20 | comment | added | Emil Jeřábek | ... $T\vdash\phi$. So, for example, this applies also to theories over intuitionistic logic (where the argument with negations does not work). | |
| Apr 14, 2023 at 16:19 | comment | added | Emil Jeřábek | If you use interpolation, you can prove conservativity directly rather than by contradiction. That is, if $T'\vdash\phi$ for a $\Sigma$-sentence $\phi$, there are (just by collecting the axioms used) $\psi_1$ in $\Sigma_1$ and $\psi_2$ in $\Sigma_2$ such that $T_1\vdash\psi_1$, $T_2\vdash\psi_2$, and $\vdash\psi_1\to(\psi_2\to\phi)$. By Craig’s interpolation, there is a $\Sigma$-sentence $\psi$ such that $\vdash\psi_1\to\psi$ and $\vdash\psi\to(\psi_2\to\phi)$. Then $T_1\vdash\psi$, thus $T\vdash\psi$ by conservativity, and likewise $T_2\vdash\psi\to\phi$ implies $T\vdash\psi\to\phi$, thus ... | |
| Apr 14, 2023 at 15:57 | comment | added | Holo | @EmilJeřábek I fixed up the wording to be more clear | |
| Apr 14, 2023 at 15:56 | history | edited | Holo | CC BY-SA 4.0 |
added 167 characters in body
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| Apr 14, 2023 at 15:41 | history | answered | Holo | CC BY-SA 4.0 |