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 and be first-order theories. If and are consistent and the intersection is complete (in the common language of and ), 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.

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