Completeness condition in Gödel first incompleteness theorem superflous

Wikipedia says:

Theory is complete if it is a maximal consistent set of sentences.

Than it says:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

But completeness definition says that theory is consistent so above definition says that

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and (maximal and consistent).

So it should be enough to say:

Any effectively generated theory capable of expressing elementary arithmetic cannot be complete.

Am I right or am I missing something?

The fact of the matter is that it is not uncommon for people to define a complete theory as a theory that for each sentence $\phi$ in its language contains (at least one of) $\phi$ or $\lnot\phi$. The only difference between this definition and Wikipedia's is that this one makes any inconsistent theory complete.
• The other version provided by wiki article is also consistent because is says that theory must contain either $\phi$ or $\neg \phi$. So for my understanding of English language it may not contain both so it is consistent. Aug 23 '13 at 8:32