 A related point holds for Frege and Russell. They possessed the conception of a hierarchy of logical levels, and in principle they, too, could have isolated first-order logic, and thus have accomplished step \((\alpha)\). But they never considered isolating the lowest level of the hierarchy as a free-standing system. There are both philosophical and mathematical reasons for this. As a philosophical matter, the logicist project aimed to show that “mathematics can be reduced to logic”: and they conceived of the entire hierarchy of types as constituting logic. And then, as a mathematical matter, second-order logic was necessary to their construction of the integers. So they had no compelling reason, either philosophical or mathematical, that would have led them to focus on the first-order fragment.