You can make it precise if the language in which you define a "useful real" is richer than the language in which individual useful reals must be defined. For in

You can make it precise if the language in which you define a "useful real" is richer than the language in which individual useful reals must be defined. For instance, model theorists will talk about definable reals in a model of set theory: https://en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC https://en.wikipedia.org/wiki/Definable_real_number#Definabi... > A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (see Kunen 1980, p. 153). This notion cannot be expressed as a formula in the language of set theory.