Full Idea
Second-order arithmetic is categorical - indeed, there is a single formula of second-order logic whose only model is the standard model ω, consisting of just the natural numbers, with all of arithmetic following. It is nevertheless incomplete.
Gist of Idea
Although second-order arithmetic is incomplete, it can fully model normal arithmetic
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.47
A Reaction
This is the main reason why second-order logic has a big fan club, despite the logic being incomplete (as well as the arithmetic).