A coherent theory is a first-order signature with a set of coherent sequents as axioms. It has a coherent syntactic category containing a conservative syntactic model. Theories of groups, rings and integral domains admit coherent axiomatizations.
Objects are coherent formulas in context and arrows are provably total, single-valued relations, modulo provable equivalence. Composition existentially eliminates the intermediate tuple. Conjunction constructs finite limits, existential formulas construct images and disjunction constructs finite unions. It is a coherent category.
The canonical model interprets each sort as its truth-context object, each operation by its term graph and each relation by its atomic subobject. A coherent formula is interpreted by its own formula-in-context object. Consequently a coherent sequent holds in this model exactly when it is derivable in the theory.
Articles by others on the same topic
There are currently no matching articles.