This has been nagging me, and might be an unfit question, but still:
I've been taught that completeness of a theory $T$ means that for any sentence $\varphi$ in the language of the theory, we have either $T \vdash \varphi$ or $T \vdash \neg\varphi$. If this property doesn't hold (so if there is some $\varphi$ that the theory says nothing about), we have an incomplete theory.
Now, copying from wikipedia, we have:
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic).
These two notions of completeness seem very far away from each other, and when I ask others that know more about this, they tell me that it's incompleteness in a different sense, and that it's too technical to cover briefly. So the question is:
Is there some difference between my understanding of incompleteness, and the one Gödel meant? And how do algorithms fit into all of this?
I want the 'philosphical' difference, not sketches of Gödel's proofs.
Thanks in advance.