 While mathematicians and Poincaré himself focused on the problems raised by the contradictions insofar as they involved the foundations of specific mathematical notions (the continuum, the natural numbers, the theory of cardinal and ordinal numbers), Russell directly attacked the comprehension principle, that is, the assumption that certain propositional functions determine a class (see also  2.3  above). The paradoxes prove that a propositional function may be well-defined for every argument, and yet the collection of the values for which it is defined need not be a class. So the crucial problem becomes logical: to give a criterion for selecting those propositional functions which give rise to classes (understood as well-defined objects).