Robinson's joint consistency theorem

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Robinson's joint consistency theorem is an important theorem of mathematical logic. It is related to Craig interpolation and Beth definability.

The classical formulation of Robinson's joint consistency theorem is as follows:

Let $T 1$ and $T 2$ be first-order theories. If $T 1$ and $T 2$ are consistent and the intersection $T_1\cap T_2$ is complete (in the common language of $T 1$ and $T 2$ ), then the union $T_1\cup T_2$ is consistent. Note that a theory is complete if it decides every formula, i.e. either $T \vdash \varphi$ or $T \vdash \neg\varphi$ .

Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:

Let $T 1$ and $T 2$ be first-order theories. If $T 1$ and $T 2$ are consistent and if there is no formula $\varphi$ in the common language of $T 1$ and $T 2$ such that $T_1 \vdash \varphi$ and $T_2 \vdash \neg\varphi$ , then the union $T_1\cup T_2$ is consistent.