 Despite these reactions to the rigor of the presentation, PM nevertheless was studied carefully by those interested in the new symbolic logic including David Hilbert and those in his school in Göttingen (see Ewald &amp; Sieg 2013: 3 and Chwistek 1912). Primarily at issue were the kinds of assumptions Whitehead and Russell needed to complete their project. Although Principia succeeded in providing detailed derivations of many major theorems in finite and transfinite arithmetic, set theory, and elementary measure theory, three axioms in particular were arguably non-logical in character: the axioms of infinity, reducibility and the “multiplicative axiom” or Axiom of Choice. The axiom of infinity in effect states that there exists an infinite number of objects. Arguably it makes the kind of assumption generally thought to be empirical rather than logical in nature. The multiplicative axiom, later added to Zermelo’s axioms as the Axiom of Choice, asserts the existence of a certain set containing one element from each member of a given set. Russell objected that without a rule guiding the choice, such an axiom was not a logical principle. The axiom of reducibility was introduced as a means of overcoming the not completely satisfactory effects of the  theory of types,  the mechanism Russell and Whitehead used to restrict the notion of a well-formed expression, thereby avoiding Russell’s paradox. Although technically feasible, many critics concluded that the axiom was simply too ad hoc to be justified philosophically. Initially at least, Leon Chwistek (1912) believed that it led to a contradiction. Kanamori sums up the sentiment of many readers: 