 We have devoted a fair amount of space in this entry to the power of second-order logic to characterize structures (see  §7.1).  Putting it briefly, if a structure has mathematical interest, it is second-order characterizable. Structures constructed by means of the Axiom of Choice may escape being second-order characterizable. But what is the power of second-order logic to demonstrate the existence of structures? The axioms, including the Comprehension Axioms and the Axioms of Choice, can be satisfied in a one element domain. So we cannot prove from the axioms that there are any structures of any size \(&gt;1\). This undermines Quine’s characterization of second-order logic as “set theory in sheep’s clothing” (1970, section heading in chapter 5), meant probably to suggest that the ontological commitments of second-order logic are on the same level as those of set theory.