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 ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ be first-order theories. If ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are consistent and the intersection ${\displaystyle T_{1}\cap T_{2}}$ is complete (in the common language of ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$), then the union ${\displaystyle T_{1}\cup T_{2}}$ is consistent. A theory ${\displaystyle T}$ is called complete if it decides every formula, meaning that for every sentence ${\displaystyle \varphi ,}$ the theory contains the sentence or its negation but not both (that is, either ${\displaystyle T\vdash \varphi }$ or ${\displaystyle T\vdash \neg \varphi }$).

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

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