Full Idea
A theory is 'decidable' iff there is a mechanical procedure for determining whether any sentence of its language can be proved.
Gist of Idea
A theory is 'decidable' if all of its sentences could be mechanically proved
Source
Peter Smith (Intro to Gödel's Theorems [2007], 03.4)
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.24
A Reaction
Note that it doesn't actually have to be proved. The theorems of the theory are all effectively decidable.