Full Idea
There are two routes to Incompleteness results. One goes via the semantic assumption that we are dealing with sound theories, using a result about what they can express. The other uses the syntactic notion of consistency, with stronger notions of proof.
Gist of Idea
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
Source
Peter Smith (Intro to Gödel's Theorems [2007], 18.1)
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.156