 For the reasons internal to type theory explained above, a Card cannot be an object within any type within the official ontology of type theory. For its would-be domain of definition would not only have to straddle distinct types, but also include classes of all types. But that is not possible for any type-theoretically admissible function or operation. This fact also precludes Russell from using Frege’s trick to ensure an infinity of  numbers.[26]  For Frege had each natural number n be the number of preceding natural numbers. For the latter to be thus numbered, they have to be objects in the official ontology—which, however, as just observed, Russell’s Card(inal)s are not. 