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.

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