Barkeep is trying to catch Bomber, but our instinct is that this cannot be done, because Bomber already knows everything Barkeep is going to do–after all, he was Barkeep, and he has a good memory. But it does not work quite like that. As we saw in Looper, the notion of the future self knowing what the past self does has an inherent flaw, which here is that the past self is reacting to the future self and so the information is changing.
Let us look at the complicated one, the thwarted bombing. In the original history there was no bombing, because Bomber did not exist; we go through quite a bit of alteration and reconstruction before we get Bomber, but eventually Bomber sets that bomb.
We see only one version of those events, although we see it from two different perspectives; there are quite a few other versions of them. As noted, in the original history there was no bomb, no explosion; in the history we see (twice), John arrives to disarm the bomb, gets in a firefight with Bomber, prevents the bomb from doing its intended damage but suffers the injuries from being caught in its incendiary blast. We ask ourselves how John knew there was a bomb there, and there is really only one possible answer: there was a history of the world in which the bomb exploded, killing some unreported number of people. In that sense, John is not preventing a bombing but rather undoing one. This is perhaps the inherent problem with police time travel stories: if the crime did not happen, it cannot be prevented, and if it did happen and you prevent it, it did not happen. Barring some application of the misunderstanding of Niven’s Law, temporal police agencies fail.
There is, though, another problem. There is probably a twenty to thirty year age difference between John and Bomber. We know that John was two weeks old in 1945 and was removed from time in 1970, making him twenty-five years old. We know that he had an illustrious career. It is not clear when the mission to prevent the bombing occurred. On the one hand it seems as if it was the first mission on which he was sent, while on the other hand it seems that after the reconstructive surgery which followed it there was much talk about how many trips he had already made and that he was about to launch his final mission. Yet the last time we see Barkeep he was certain he would never become Bomber, and Bomber had a significant career, probably spanning quite a few years itself. The minimum gap we could imagine is about five years, and twenty is considerably more plausible. Let us settle for ten, for the sake of discussion.
Bomber sets the bomb in 1970, and it destroys whatever was targeted. No one arrives to stop it. Then at some point John is detailed to attempt to prevent this bombing, and he arrives. There is no reason to suppose that Bomber waited for him, and therefore John will successfully disarm the bomb. The Bomber who planted this bomb did not know John would do this, because he never was that version of John. It will take this John ten years to become that Bomber–then that Bomber will know that that John defused that bomb, and will stay to prevent that. He will get in a gun battle with John, and flee just before the bomb detonates, giving John sufficient time to remove it from the target but insufficient to contain it. John is caught in the blast, and badly burned. He finds his violin case, and returns to headquarters.
Note that Barkeep is not there. Barkeep has no reason to be there–in his memory, he removed the bomb and contained it, and there was no sign of Barkeep. It will be a couple of years before the version of John who remembers being caught in the blast becomes Barkeep, and makes the trip to attempt to catch Bomber at the scene. That is also when Barkeep gives the violin case to John, changing another detail–but it must be a token gesture, because John must have reached it on his own in order for Barkeep to be there at all. That means that that Barkeep will now become the future Bomber, who will also remember that Barkeep was there.
Note that this is unlike Next, in which Cadillac’s knowledge enabled him to dodge bullets and blows and evade pursuit: Bomber does not know what will happen, only what happened when he was that age, and as he changes his actions in response to his memory, his younger selves change what they do to counter the new actions.
It is remarkable that he does not get caught–but perhaps he does, because in the next iteration of history he would know what he did wrong and be able to avoid making that mistake again. On the other hand, doing it right erases the knowledge of what happened when he did it wrong, so ultimately it has to stabilize into one version of events that are good enough, the version we see.