 Skolem’s remarks about first-order logic require careful interpretation (see, e.g., Ferreirós 2001: 470–74), but clearly must be viewed against the backdrop of the Grundlagenkrise of the 1920s, and of the debates between Hilbert, Brouwer, and Weyl. There are two broad tendencies within logic during these years, and they pull in opposite directions. One tendency is towards pruning down logical and mathematical systems so as to accommodate the criticisms of Brouwer and his followers. The aim was to avoid the paradoxes, to delimit the territory of “legitimate” mathematics, and to place it on secure foundations. Set theory was in dispute, and Skolem explicitly presented his 1922 results as a critique of set theoretical foundations. Weyl already in 1910 had been led by his examination of Zermelo’s system to formulate a set of logical principles that in retrospect (and despite the idiosyncratic notation) can be seen to be a form of first-order logic. In general, both Weyl and Skolem inclined, on methodological grounds, towards some sort of constructivism as a means of avoiding the paradoxes; and this meant that they viewed quantification over, say, the totality of subsets of an infinite set as something to be avoided: whatever one’s grasp on the notion of “all integers”, the notion of “all properties of integers” was far less firm. To put the matter slightly differently: the very point of axiomatizing set theory was to state its philosophically problematic assumptions in such a way that one could clearly see what they came to. But this aim would be compromised if one already presupposed in the background logic the problematical notion of “all subsets” that one was attempting to elucidate. One possibility was to restrict oneself to first-order logic; another, to adopt some sort of predicative higher-order system.