 Wittgenstein’s intermediate critique of transfinite set theory (hereafter “set theory”) has two main components: (1) his discussion of the intension-extension distinction, and (2) his criticism of non-denumerability as cardinality. Late in the middle period, Wittgenstein seems to become more aware of the unbearable conflict between his strong formalism (PG 334) and his denigration of set theory as a purely formal, non-mathematical calculus (Rodych 1997: 217–219), which, as we shall see in  Section 3.5,  leads to the use of an extra-mathematical application criterion to demarcate transfinite set theory (and other purely formal sign-games) from mathematical calculi.