The crucial difference between Church and Curry is that Curry allows each term to be ascribed a set of types - perhaps 0, 1 or many - whereas Church insists on

The crucial difference between Church and Curry is that Curry allows each term to be ascribed a set of types - perhaps 0, 1 or many - whereas Church insists on each term having exactly one (often annotated). Both are systems for proving type correctness of terms.