Completeness and Incompleteness Through browsing questions asked in the past about Godel's completeness and imcompleteness theorems, I've come to see that the sense of completeness in both of the theorems are different. However, I can't see how they are different! 
Does this distinction boil down to that of truth and proof? Is it the case that the completeness theorem talks about the relationship between truth and proof, while the incompleteness theorems only relate to provability?
Could someone please elaborate on the sense in which both notions of completeness are different? 

To put it real short: Godel's completeness result is about (first-order) logic itself, and his incompleteness result is about arithmetic.
One way to put the results is: we can axiomatize all of (first-order) logic itself, but we cannot axiomatize all of arithmetic.
