  Notice that the sentence σ2 contains an infinite quantifier: it expresses the essentially second-order assertion that every countable subset has a least member. It can in fact be shown that the presence of this infinite quantifier is essential: the class of well-ordered structures cannot be characterized in any finite-quantifier language. This example indicates that infinite-quantifier languages such as  L(ω1,ω1) behave rather like second-order languages; we shall see that they share the latters' defects (incompleteness) as well as some of their advantages (strong expressive power). 