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Full Idea
Going second-order in arithmetic enables us to prove new first-order arithmetical sentences that we couldn't prove before.
Gist of Idea
Second-order arithmetic can prove new sentences of first-order
Source
Peter Smith (Intro to Gödel's Theorems [2007], 23.4)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.206
A Reaction
The wages of Satan, perhaps. We can prove things about objects by proving things about their properties and sets and functions. Smith says this fact goes all the way up the hierarchy.