<i>Closed timelike curves are intrinsic and irreducible features of Gödel space-times.12 If they are possible, so is time travel. And with time travel, certain paradoxes arise. Imagine a traveler arriving in the past and killing his own grandfather.13 Would he survive the encounter if he broke the causal chain leading to his own existence?<p>There are two popular views about how these kinds of paradox might be managed. The first is committed to an ensemble of equally concrete but different versions of the physical world. Travelers into the past arrive in worlds that are distinct from those that they left. They are free to kill their grandfather secure in the knowledge that their grandfather is not really their grandfather, but something like his counterpart. David Deutsch and Michael Lockwood think that restrictions posed by classical systems on the actions of time travelers imply that time travel must must displace travelers into different worlds.14 It is by no means clear that time travelers under such a scheme are really following a closed timelike curve. Closed timelike curves are paths that return to, or very close to, their own spatiotemporal starting points.15<p>On quite another view, time travel really does return an agent to his very own past. A temporal rerun is possible only if everything he does in the past is already in place in his history. This means everything. Consistency might be maintained through the most ordinary of physical processes: the gun misfires, or the bullet dribbles out inconclusively, or at the very last moment your grandfather ducks to tie his shoelaces. Your efforts can make the past what it was but they cannot make the past different from what it was.</i><p>There is a third, which is that you would arrive in the past only to find that you had no agency at all, no free will. In this formulation you would never attempt to kill your grandfather, and no faulty guns or lucky grandparents are needed. If the present is a hypersurface sweeping through spacetime, leaving the past in its wake, and the past is a set structure... you’re an automaton if you travel to the past. Presumably in this kind of natural order, the future doesn’t exist except as a word and concept. There is the fixed past, and the hypersurface of the present.<p><i>Or</i> the future and past are both fixed, and the hypersurface of the present is some strange artifact we perceive. Or... who knows?
Slightly off-topic but after reading Douglas Hofstadter's books the one thing that I never managed to grok was Godel's incompleteness theorem. It's implications are fascinating but my math background never allowed me to completely understand it's proof.