The Futurama film Bender's Big Score treats doppelgangers in an unusual fashion, worthy of consideration despite its humorous application. We are told that the code is self-correcting, that paradox is automatically prevented; we are very quickly given an example of this prevention at work. One of the aliens creates a temporal duplicate of himself by traveling back one day, and then showing up with himself seconds after his own departure. Within minutes, a large piece of equipment falls on one of the pair, killing him instantly.
The film is consistent in one significant aspect of this. When a character travels to the past, he duplicates himself; if, however, the version of him that has not traveled to the past does not do so at the right moment, that character begins collecting "doom", which aggregates to the point that he meets with a fatal accident. What is consistent is that the one who meets with the accident is always the one who did not travel to the past; he is treated as the duplicate, and eliminated. This is how the "code" prevents paradox.
Under fixed time theory this is impossible. If you arrived in the past, then you departed from the future, and nothing can prevent you from doing so.
Parallel dimension theory faces an entirely different problem. There is no reason why the alien could not travel to the past, meet himself, and keep himself from traveling to the past; there would then be two of him in that universe, and none in the one from which he departed--but this is not a paradox nor a problem under this theory, merely the common result of such travel between dimensions. It would be much the same for divergent dimension theory, in which the departure of the second version is more problematic than his continued presence. No correction is necessary.
Yet there remains a problem under these theories: as the two aliens enter the room, they should encounter a group who are completely unaware that one of them just left for the past. He did not do so in this universe; he did so in a different universe, of which these people have no memory. Further, in that other universe, the alien stepped into the time sphere and vanished, never to be seen again; it is unlikely that another time travel experiment was attempted there.
Of the major time travel theories this leaves replacement theory. Here, too, though, we encounter insurmountable obstacles: in order for the alien to arrive in the past he must depart from the future; if he fails to do so we have a paradox. But the paradox is not that the alien who did not make the trip cannot exist; it is that the one who made the trip cannot exist. By preventing his own departure from the future, the time traveling alien has effectively killed his own grandfather, that is, undone the basis for his own existence in the past. We have a classic infinity loop. If the code were truly self-correcting in a replacement theory universe, then either the alien could not have arrived in the past or he could not have prevented himself from departing from the future--in which case the effect of the code makes the nature of the universe indistinguishable from fixed time.
So you cannot repair a doppelganger problem by killing the non-time-traveling version. If a temporal duplicate is a paradox (which it is not under some theories of time), then it is the one who made the trip who is the paradox, because he has no origin in the history that was.
For the sake of analysis, we will overlook this fatal error in time and continue in our next article with a look at Bender's archaeological thieving spree.
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