Timeline for Is the union of two conservative extensions of a theory conservative?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2023 at 15:47 | vote | accept | Giacomo Cozzi | ||
| Apr 15, 2023 at 8:50 | comment | added | Emil Jeřábek | @GiacomoCozzi No, the joint consistency theorem is applied only once. $T_1\cup T'$ and $T_2\cup T'$ are consistent because of the conservativity of $T_1$ and $T_2$ over $T$. | |
| Apr 15, 2023 at 7:44 | comment | added | Giacomo Cozzi | If I understand correctly, the joint consistency theorem is being applied three times to show that $T_1 \cup T'$ , $T_2 \cup T'$ and then finally $T_1 \cup T_2 \cup T'$ are consistent. But how do we know $T_1 \cap T'$ / $(T_1 \cup T_2) \cap T'$ are complete? Perhaps this is a simple fact on intersections with general complete theories I do not know? | |
| Apr 14, 2023 at 15:41 | comment | added | Holo | Interesting, the proof I know for the joint consistency theorem is to show the conservative variation using Craig’s interpolation (see my answer) and from there the standard variation is immediate, I guess that the difference comes to preference of using models or being purely syntactic | |
| Apr 14, 2023 at 15:31 | comment | added | Emil Jeřábek | ... using the cut elimination theorem. (More common proofs of Craig’s interpolation theorem derive it from the joint consistency theorem.) In this way you prove whatever version of the joint consistency theorem you want with essentially the same effort. | |
| Apr 14, 2023 at 15:29 | comment | added | Emil Jeřábek | @Holo I don’t think I’ve ever seen it proved in that way. The most common proof of the joint consistency theorem is to derive it from the joint consistency lemma (if $A$ and $B$ are structures such that $A\restriction\Sigma\equiv B\restriction\Sigma$, then there are elementary extensions $A'\succeq A$ and $B'\succeq B$ such that $A'\restriction\Sigma\simeq B'\restriction\Sigma$). This naturally proves the version with $T$ complete. Another proof is to derive it from Craig’s interpolation theorem, which in turn can be proved (in a sequent form) by induction on the length of a cut-free proof ... | |
| Apr 14, 2023 at 15:21 | comment | added | Holo | Usually the "common formulation" of the joint consistency theorem is proven using the Conservative Extension formulation, I will soon write an answer that proves the variation without assuming the Robinson’s joint consistency theorem | |
| Apr 14, 2023 at 15:17 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| Apr 14, 2023 at 15:08 | history | answered | Emil Jeřábek | CC BY-SA 4.0 |