I see what I was missing; I was considering "complete" to be a statement about a set of axioms, not about a structure.
Originally posted by Hurkyl
Either you are remembering incorrectly or he is talking about a different kind of incompleteness (unrelated to the meaning of incompleteness in Godel's theorem).
.Hurkyl wroteYou may find it interesting to reflect upon the fact that euclidean geometry is both consistent and complete.
the completeness of algebra which was itself equivalent to the completeness of the natural numbers