more on this theme | more from this thinker | more from this text
Full Idea
The theorems of any properly axiomatized theory can be effectively enumerated. However, the truths of any sufficiently expressive arithmetic can't be effectively enumerated. Hence the theorems and truths of arithmetic cannot be the same.
Clarification
In an expressive arithmetic we can enumerate theorems, but not truths
Gist of Idea
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent)
Source
Peter Smith (Intro to Gödel's Theorems [2007], 05 Intro)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.37