How could Joe end up in the field?

I guess this is more a question about the nature of time travel more than anything else.

At the end of the movie, Joe shoots himself in a field and changes the course of history. What I'm curious about is how Joe ended up in the field given the change in history.

If his future self had never existed he'd never have gone on the mission he did and thus he never would have ended up in the field. It seems as though once he'd killed himself, he couldn't have killed himself because what led to that event was the existence of his future self. So theoretically, time and space should have just folded in on itself and popped out of existence.

What do you think? Is there something I'm missing that would explain a dead Joe in a cornfield and how we would have ended up there without their having been a future version of himself to try and stop?