Full Idea
First Incompleteness Theorem: any properly axiomatised and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix.
Gist of Idea
First Incompleteness: arithmetic must always be incomplete
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.2
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.5
A Reaction
This is because it is always possible to formulate a well-formed sentence which is not provable within the theory.