Timeline for Interesting meta-meta-mathematical theorems?
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| Jul 27, 2014 at 5:57 | comment | added | Ioachim Drugus | Good points. I was aware of more general statement with arbitrary T and T', but I phrased my statement so that the correlation "meta" between T and T' helps resolve the difficulties you discribed. A metatheory is said to be a theory "about" a theory. I treat this aboutness as a provision that both theories can be replaced by equivalent theories, such that difficulty with cardinality does not occur. Also, T' must encrypt the "knowledge" in T, and to avoid the difficulty in your last sentence, we add the missing correlations between symbols of two theories after taking their union. | |
| Jul 27, 2014 at 4:38 | comment | added | Andreas Blass | This is not strictly true because $T$ might have only finite models of a fixed size and $T'$ might have only infinite models. If we exclude that case, then any two theories, in disjoint languages, that both have infinite models have a common conservative extension, namely their disjoint union (and its consequences if you require theories to be closed under deduction). So this has nothing to do with "meta". Also, because you made the languages disjoint, $T'$ won't be able to prove anything nontrivial in the language of $T$. | |
| Jul 27, 2014 at 1:48 | history | answered | Ioachim Drugus | CC BY-SA 3.0 |