For the preceding natural numbers to be numbered by Frege's trick, those natural numbers must be objects in the official ontology.

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 num