I'm currently trying to understand the notion of a naked singularity. After consulting books by Wald and Choquet-Bruhat, it seems that for a naked singularity one must have that the causal curves can leave the singularity and extend to infinity (that is, they will not hit the horizon). This looks like the notion of "geodesically complete" to me. However, doesn't the very fact that we have a naked singularity show that the spacetime is geodesically incomplete, i.e. the timeline geodesics cannot be infinitely extended?
I'm extremely confused because it seems that the spacetime is then geodesically complete and incomplete at the same time.
I would be very grateful if anyone could clarify this.