Concerning mathematics, an example of an impredicative definition is the smallest number in a set, which is formally defined as : = min ( ) } } if and only if for all elements } } of } }, } } is less than or equal to } }, and } } is in } }.
12.
Thus for type 1, properties defined without mentioning any totality belong to " order 0 ", and properties defined using the totality of properties of a given order belong to the next higher order . . . . But this separation into orders makes it impossible to construct the familiar analysis, which we saw above contains impredicative definitions.
13.
Kleene observes that already Russell has set himself up with an impredicative definition that he will have to resolve, or otherwise he will be confronted with his Russell paradox . " Here instead we presuppose the totality of all properties of cardinal numbers, as existing in logic, prior to the definition of the natural number sequence " ( Kleene 1952 : 44 ).
14.
In the next paragraphs he discusses Weyl's attempt in his 1918 " Das Kontinuum " ( " The Continuum " ) to eliminate impredicative definitions and his failure to retain the " theorem that an arbitrary non-empty set of real numbers having an upper bound has a least upper bound ( cf . also Weyl 1919 ) ".
15.
After he discovered the paradox in Frege's " Begriffsschrift " he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a " doctrine of types "; see more about the unit class, the problem of impredicative definitions and Russell's " vicious circle principle " below.
16.
This is an example of the paradoxes that result from an impredicative definition that is, when an object m has a property P, but the object m is defined in terms of property P . The best advice for a rhetorician or one involved in deductive analysis is avoid impredicative definitions but at the same time be on the lookout for them because they can indeed create paradoxes.
