A comparison of HOL-ST and Isabelle/ZF

The use of higher order logic (simple type theory) is often limited by its restrictive type system. Set theory allows many constructions on sets that are not possible on types in higher order logic. This paper presents a comparison of two theorem provers supporting set theory, namely HOL-ST and Isabelle/ZF, based on a formalization of the inverse limit construction of domain theory; this construction cannot be formalized in higher order logic directly. We argue that whilst the combination of higher order logic and set theory in HOL-ST has advantages over the first order set theory in Isabelle/ZF, the proof infrastructure of Isabelle/ZF has better support for set theory proofs than HOL-ST. Proofs in Isabelle/ZF are both considerably shorter and easier to write.