Robinson's joint consistency theorem

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. A theory $$T$$ is called complete if it decides every formula, meaning that for every sentence $$\varphi,$$ the theory contains the sentence or its negation but not both (that is, 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.