Concepts exressible in infinite logic In the language of set theory the following statement expresses foundation Unlike the axiom of foundation this statement admits no non standard interpretations The concept of well foundedness can only be expressed in a logic which allows infinitely many quantifiers in an individual statement As a consequence many theories including peano arithmetic which can not be properly axiomatised in finite logic can be in a suitable infinite logic Complete infinite logics Two infinite logics stand out in their completeness These are and The former is standard finite first order logic and the latter is an infinite logic which only allows statements of countable size