 With this turn to the ramified theory of types, along with the extra axioms of reducibility, and infinity, it is possible for PM to define a version of Frege’s construction of the Natural Numbers so that the “Peano axioms” can be proved from logic alone. This takes up to section ∗120, well into Volume II. At this point the alternative to “Frege’s Theorem” is completed, in the sense that we are presented with a consistent development of the natural numbers, based on a theory of higher-order logic with a number of additional axioms. Philosophers soon followed Ludwig Wittgenstein (1922) and disputed the idea that these additional axioms, the axioms of reducibility and infinity, are really logical truths, and so denied that the logicist program of reducing arithmetic to logic was any more successful than Frege’s attempt had been.