 And fall it did. Frege, it turned out, had gone in for overkill with the formal system that was to vindicate his logicism. He sought to unify all of arithmetic and analysis within a general theory of classes, or extensions (of concepts). Classes were supposed to be logical objects par excellence. The strategy was to define the natural numbers, say, as particular classes within a much more capacious universe of abstract, logical objects. Using the definitions, one would then derive the first principles of arithmetic (the Dedekind-Peano axioms, say) as theorems within the theory of classes. To that end one would exploit, ultimately, only the deeper underlying axioms (or basic laws) governing classes themselves. For more details on this strategy, see  §1.2.4.  