Set Theory, Higher Order Logic or Both?

The majority of general purpose mechanised proof assistants support versions of typed higher order logic, even though set theory is the standard foundation for mathematics. For many applications higher order logic works well and provides, for speciication, the beneets of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientiic or engineering background already know set theory, but not higher order logic. This paper discusses some approaches to getting the best of both worlds: the expressiveness and standardness of set theory with the eecient treatment of functions provided by typed higher order logic.