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 eleme