Full Idea
By Gödel's First Incompleteness Theorem, there cannot be a negation-complete set theory.
Gist of Idea
There cannot be a set theory which is complete
Source
Peter Smith (Intro to Gödel's Theorems [2007], 01.3)
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.5
A Reaction
This means that we can never prove all the truths of a system of set theory.