Lindebaum's Lemma seemingly inconsistent with Gödel's incompleteness theorem? Lindenbaum's Lemma: Any consistent first order theory $K$ has a consistent complete extension.
First Incompleteness Theorem: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.
If the hypothesis of the First Incompleteness Theorem holds for a theory $K$, why doesn't an application of Lindenbaum then yield a contradiction? 

Why is the completion of an effectively generated theory stays an effectively generated theory?
In fact, this shows that it doesn't.
