nLab Robinson consistency theorem

Statement

Proposition

Robinson’s Joint Consistency Theorem: If $T_1$ is a consistent $L_1$-theory, and $T_2$ is a consistent $L_2$-theory, then their union $T_1 \cup T_2$ is consistent when their intersection $T_1 \cap T_2$ is complete as an $(L_1 \cap L_2)$-theory.

Proof

Apply the Craig interpolation theorem.

This may be used to demonstrate the Beth definability theorem.

References

• Daniele Mundici, Robinson’s Consistency Theorem in Soft Model Theory, Transactions of the American Mathematical Society 263 1 (1981) 231-241 [doi:10.2307/1998653, jstor:1998653]

• Daniel Găină, Andrei Popescu, An Institution-Independent Proof of the Robinson Consistency Theorem, Studia Logica: An International Journal for Symbolic Logic, 85 1 (2007) 41-73 [jstor:40210759]