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 and be first-order theories. If and are consistent and the intersection is complete, then the union is consistent. Note that a theory is complete if it decides every formula, i.e. either or .
Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:
Let and be first-order theories. If and are consistent and if there is no formula in the common language of and such that and , then the union is consistent.
