TITLE: What is the difference between Gödel's completeness and incompleteness theorems?
QUESTION [26 upvotes]: What is the difference between Gödel's completeness and incompleteness theorems?

REPLY [11 votes]: I'll add some comments...
It is useful to state Gödel's Completeness Theorem in this form : 

if a wff $A$ of a first-order theory $T$ is logically implied by the axioms of $T$, then it is provable in $T$, where "$T$ logically implies $A$" means that $A$ is true in every model of $T$.

The problem is that most of first-order mathematical theories have more than one model; in particular, this happens for $\mathsf {PA}$ and related systems (to which Gödel's (First) Incompleteness Theorem applies).
When we "see" (with insight) that the unprovable formula of Gödel's Incompleteness Theorem is true, we refer to our "natural reading" of it in the intended interpretation of $\mathsf {PA}$ (the structure consisting of the natural number with addition and multiplication). 
So, there exist some "unintended interpretation" that is also a model of $\mathsf {PA}$ in which the aforesaid formula is not true. This in turn implies that the unprovable formula isn't logically implied by the axioms of $\mathsf {PA}$.