Full Idea
The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures.
Clarification
For 'categorical' see Idea 13636
Gist of Idea
The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.80
A Reaction
So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project.
Related Idea
Idea 13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]