 Boolos (1987), after raising qualms about the universal number, provided an ingenious model (which had been anticipated informally by Geach (1975: 446–7)) to allay the misgiving about the consistency of full second-order logic with HP (the system now known as FA, for ‘Frege Arithmetic’). Simply take the natural numbers along with the distinct object ω as the elements of the domain. The element ω serves as the denotation of any term of the form #xΦ(x) where Φ is satisfied by infinitely many elements. Note, however, that this consistency proof works only when FA is taken on its  own.[38]  The Geach–Boolos model cannot be relied upon to secure the consistency of FA in conjunction with other theories, such as set theory, that one might wish to extend with FA. Since, however, counting finite extensions ought to be a universally applicable intellectual operation, regardless of subject matter, it will be the exception rather than the rule that FA is to be applied only to the natural numbers (plus, perhaps, the unnatural factotum ω). Rather, FA should be applicable not only to concrete objects, but also to abstract mathematical entities such as real numbers and sets. Provided only that one has a criterion of identity for the objects in question, one should be in a position to count any finite collection of them. 