On sets, types, fixed points, and checkerboards

Abstract

Most current research on automated theorem proving is concerned with proving theorems of first-order logic. We discuss two examples which illustrate the advantages of using higher-order logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathematics than axiomatic set theory. Even theorems of first-order logic can sometimes be proven more expeditiously in higher-order logic than in first-order logic. We also note the need to develop automatic theorem-proving methods which may produce proofs which do not have the subformula property.