Answer by Holo for Is the union of two conservative extensions of a theory...
As Emil pointed out, assuming $\Sigma=\Sigma_1\cap\Sigma_2$, the union of 2 such theories is always conservative extension, Emil had proven it using the standard formulation Robinson’s joint...
View ArticleAnswer by Emil Jeřábek for Is the union of two conservative extensions of a...
Yes, this is true (and somewhat nontrivial). That is, if $T$ is a theory in a language $\Sigma$, and $T_1$ and $T_2$ are conservative extensions of $T$ in languages $\Sigma_1$ and $\Sigma_2$...
View ArticleAnswer by Arno for Is the union of two conservative extensions of a theory...
The answer is no. Let $\phi$ be a non-theorem in the language of $T$, and let $c,d$ be two fresh symbols for constants. Now we can take $T_1$ to add the axiom $(c = d) \Rightarrow \phi$, and $T_2$ to...
View ArticleIs the union of two conservative extensions of a theory conservative?
Background/MotivationA theory T over a signature(language) Σ is a set of formulae over Σ. These formulae are called the non-logical axioms of T.To talk about what is provable in T we can agree on...
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