. 

 ==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 cannot 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