Given the general irreverence of Futurama, one might expect that a movie based on the animated television series would not take anything seriously--and indeed, Bender's Big Score, the direct to video feature-length cartoon based on the series makes comedic reference to The Matrix (the means of altering the past involves reciting a line of computer code), Terminator (when the robot is sent to kill someone in the past, he dons a pair of sunglasses, asserting that it is "bright" in the past), Al Gore (seen in 2000 talking about the technological advances he will bring when he is President), and a not-subtle swipe at the television network that cancelled the program.  Yet there are serious temporal elements in the film worth discussion, and a broad fanbase that might be interested in whether the story "works" under any theory of time.

Before there is any time travel, we are introduced to aliens called Scammers who have the ability to find information that can be used for identity theft, who manage to steal Earth and everything on it.  Then they discover the secret of time travel, tattooed in a somewhat inconvenient location of main character Fry's anatomy.  This gives them the opportunity to rob history of its treasures.  However, the code is unidirectional:  the traveler can go to the past, but cannot return to the future.

Their solution is to send Bender the robot on various missions to rob the past, and for him to hide with the stolen merchandise in the caves which conveniently happen to be unexplored beneath the office, until the logical moment for his return.  Over objections that this might bring the end of everything, the alien scammers send the robot back repeatedly, gathering such treasures as the Mona Lisa and the Colonel's Secret Recipe.

Once the scammers are satisfied that history, too, has been fully pillaged, they determine to eliminate the code by erasing it from Bender's memory and killing Fry; but Fry has learned how to read and use the code, and so escapes to the past, to moments after he fell into the cryogenic tube which brought him Buck Rogers style to the future originally.  Bender is sent to terminate him; but Fry incidentally uses the code again so that there are two of him before Bender arrives.  There ensues a merry chase in which Fry and Bender each temporally duplicates himself.

In a significant sub-plot, administrator Hermes loses his head--quite literally having it severed from his body in a serious accident.  It is preserved in a jar for re-attachment to his body, but the interference of the Scammers delays the procedure.  Afraid that he will lose his gorgeous wife to a gorgeous superstar, he has Bender steal his own body from his past self so that it can be reattached in the present.

The scammers are ultimately defeated (without the use of time travel), and earth reclaimed, and in the end Bender travels from the end of the story to the beginning, to tattoo the code on a place on Fry's body where it will remain unnoticed until a trip to the nude bathing planet.  That, then, is where the time travel story begins:  the last trip might be the first trip.  We will begin there--next time.

The time travel story might begin at the end, because it is in the last minutes of the movie that Bender leaves to place the tattoo on Fry; time travel is initiated in the film when that tattoo is discovered to contain the time travel code.  No one can travel to the past until the code is discovered, and the code cannot be discovered until Bender brings it to the past.  In this form of predestination paradox we have information being passed from the future to the past, in essence knowledge known in the future becoming the basis for its own presence in the past, and thus for its own existence in the future.  That is, Bender can take the code and place it on Fry because he found it on Fry initially.  As elsewhere discussed, what matters is that the knowledge is available in the future, and thus can be taken to the past.  We saw similar events in connection with Mr. Scott's transwarp teleportation formula in the recent 2009 film Star Trek, and in connection with transparent aluminum in Star Trek IV:  The Voyage Home.

The problem in the Futurama version is that the original source of the knowledge has been lost, and there are few if any clues to enable us to reconstruct a history of the universe which would have led to the discovery of the code and its placement on Fry in the past.

It is almost certain that in the original history it was Bender who left from the future to carry the code; that tells us that the code came into the possession of the Planet Express Delivery Company.  The question is, how did that happen?  The answers might be

1. they found it themselves somewhere, and decided to send it to themselves in the past;
2. the Scammer aliens found it and gave it to Bender to use, who in some way was able to save it for the future;
3. someone else found it and decided to give it to them so that they could use it to alter the past.

These answers raise the question of when and where they found it.  If it was found early enough, the Scammers could have used it as they do in the timeline we observe.  Fry would not have the code, so his life would never be threatened and his doppelganger would not exist.  If, though, it was found later--perhaps after humanity had been exiled to Neptune--then the Scammers never had it until after Bender attached it to Fry.  That would mean that they did not rob history, and that among all the duplicate Benders which congregated in the cave there was always one in a tuxedo, because his was the first trip.  He, however, has no memory of the other trips, which he did not take.

Time travel in the film creates several impossibilities; it seems best to assume that there was no time travel in the original history up to the point at which Bender takes the code back to Fry, or that point could not have been reached.  This means the Scammers never had it, so whoever found it gave it to the Planet Express Delivery Company.  Since quite independent of time travel, that company was the point of initial attack, it may have been thought that they could use time travel to prevent themselves from falling prey to the scams which led ultimately to the taking of Earth.  The decision to deliver it to themselves via Fry's anatomy has merit in that they might hope their past selves would identify the use of the code before the Scammers attacked, and could use it against the invaders.

Thus the most plausible reconstruction of the original timeline has the scammers taking Earth, ultimately shipping humanity to Neptune; humanity regaining their planet by invasion; then the time code being discovered and given to Bender, who departs to attach it to Fry, creating the altered history in which Fry has a tattoo.

This, then, sets up the time travel that we see in the film.

The movie 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 with a look at Bender's archaeological thieving spree.

The titular robot goes on a thieving spree through history, at the direction of three Scammer aliens stealing everything of value throughout all time.  For his first theft, he wrestles the Mona Lisa from Leonardo da Vinci before the painting is finished, reportedly leaving the famed artist at least critically wounded.

This unfinished painting points up an aspect of historic treasures which should be recognized.  The painting Bender stole is almost worthless--at most, worth considerably less than the version known to us.  Before it was completed, it was just an unfinished canvas.  It might have some value now as an unfinished masterpiece by a master--if the da Vinci who now is known only for The Last Supper (which contrary to the suggestion of Bender's quip was painted prior to this work) is famed enough, an unfinished painting would be of historical interest.  Yet this still overlooks the principle aspect of what gives artistic masterpieces value.  We value great works of art because we value them; we recognize within them something great, and so attribute to them great value in our desire to possess them.  It would not be enough for the Mona Lisa to have been finished; it would also have to have been released to the world in some way, such that it could gain value by the desire of others to own it.  A work that never met public eye would never have that kind of value; its value would be purely aesthetic (and our aliens show no appreciation for any kind of value that is not monetary) or limited to the value of a previously unknown painting by a known artist.  The best the Scammers can hope is that they can proclaim it as the mysterious lost or stolen painting on which the great artist was working shortly before his death; even in this, we have to wonder whether if only one of da Vinci's two surviving paintings survived he would have been so well known, particularly as the other is part of a wall and cannot be moved.

Complicating this further, once Bender returns with the painting, having rendered it worthless and unknown in history he has eliminated any motivation he might have had for stealing it originally.  He could not have known of its existence had it not become famous; he has prevented it from becoming famous.  In so doing, he creates an(other) infinity loop, as history vacillates between the world in which Bender is motivated to steal the most famous painting which ever existed and the one in which it never did exist.

This complicates his task, in that he must wait for some objects to become famous and thus valuable before he can steal them.  It does not apply to everything he takes--his theft of Egyptian artifacts will start our subject next time, but they have value partly from their materials and partly from their great antiquity.  Yet this one theft underscores the complication facing many of the others, that of knowing when and where to steal an object for maximum value in the future.  Take them too soon, and at least some will have less value than the unfinished Mona Lisa, and all reason for having taken them at all will be erased.

The second larcenous trip Bender makes to the past is to steal the contents of an Egyptian tomb before they are interred.  This does not face the same problems as the previously considered stealing of the Mona Lisa.  Since they will spend four thousand years in the limestone caves beneath the office, they will certainly age sufficiently to pass for originals.

That, though, might be a flaw in the plan.

The treasures of King Tut's tomb were so well preserved after thousands of years in large part because of the combination of the dry climate of Egypt and the securely sealed tomb.  The conditions under which objects are stored play a significant role in their rate of deterioration; a great deal of money is spent on climate control in museums of all types, not primarily for the comfort of the patrons.  Bender's plan is to capture objects centuries previous and keep them safe in a limestone cave--one of the wetter environments of the earth.  It is certainly questionable how many will survive the thousands of years of such exposure.

Of more interest from a time travel perspective is the experience of being in that cave.  If, as suggested, the last departure was the first trip made, life in that cave would be interesting to observe, from an atemporal position.

Bender, in a tuxedo, travels from the end of the story to January 1, 2000, places the tattoo on Fry, and then enters the empty cavern to await the proper time for his return.

Before that time arrives, Bender not yet having made that trip makes his first trip to steal the Mona Lisa.  He, too, enters an empty cave; but before he leaves, a version of himself in a tuxedo arrives.  Neither Bender knows anything about the other; it is not part of the history he remembers.

Bender leaves the cave with the Mona Lisa and then goes to Egypt; in around 1351 B.C. he enters the empty cave.  Almost three millennia later he is joined by a Bender he remembers being, carrying the Mona Lisa.  Later still they are joined by a Bender in a Tuxedo, but this Bender neither recognizes them nor is recognized by them, apart from that they know it is him.  He is in their future, but they are not in his past.  Ultimately the Mona Lisa Bender leaves the cave, and moments later the Egypt Bender.  The Mona Lisa Bender must then make the trip to become the Egypt Bender, and the Egypt Bender in turn makes his next trip.

The cave continues to fill like this, each additional trip adding a Bender with a treasure, none of them knowing who the tuxedo Bender is, the tuxedo Bender not knowing what they are doing.

The question is, how much information do they exchange?  The very fact that they see each other tells them much, but for tuxedo Bender the fact that his doppelgangers are stealing for the Scammers is new information.  It is difficult to imagine loud-mouthed Bender not bragging a thousand times over about what he is doing and why.  But this interaction of the doppelgangers will impact events.  For example, if the Bender who steals the 2308 Nobel Peace Prize brags about it, the Benders who have not yet done so will gain information that will refine their own approach to that heist.  This creates multiple overlapping sawtooth snaps as the information trickles through from future to past versions of Bender ("I heard me talking about how he stole the Gutenberg Bible, and I thought that it would be easier to do it this way, so I did, but now I think that I should have tried this other idea instead").

More complicated is the impact of tuxedo Bender on the others; but this, too, is complicated by the addition of terminator Bender.

There is one anomaly in which cannot be resolved even in concept under any known model of time, given the facts of the story.  It pertains to Hermes Conrad.

Before much of anything has happened in the film, Hermes loses his head in a freak accident involving a sword falling from a wall and decapitating him.  This being a thousand years in the future, the head is preserved in a jar for reattachment to the body; but the procedure is delayed when the Scammer aliens deprive the Head Museum of its assets.  Hermes is quite concerned that his gorgeous wife will leave him for Barbados Slim, on the pretext that "boy need a father".  Indeed, Hermes' hold on this woman is weak.

As the reattachment procedure is further delayed, Hermes devises a solution, and gets Bender to assist.  He needs a body now; he had one in the past.  Bender travels to the past, kills Hermes before the accident, and brings Hermes' body, minus the head, forward to the present, where Dr. Zoidberg improperly but effectively attaches the present head to the past body.

The film treats this as if the body were like any other temporal duplicate.  The Futurama treatment of doppelgangers has already been found wanting, but in this case the problem is more complicated.  No part of Hermes ever traveled through time; the time code does not permit travel to the future, even connected with travel to the past.  Bender thus killed Hermes before his accident, took the body into the cave for an unspecified length of time which had to be at least several months, and then delivered it to Zoidberg.  However, while Bender was in the cave, events were being formed in the world above--and Hermes (being already dead) was not decapitated at the limbo contest, so his head is not awaiting a body.

For fixed time advocates this is absurd, and never could happen; Hermes was not killed in the past.  Under replacement theory it creates a simple infinity loop, since once Bender kills Hermes in the past, Hermes' head is not there in the future to ask him to do so.  Divergent dimension theory provides a simple explanation, that Hermes was now killed in the past and his non-existence in the future simply means that Bender wasted a trip; there is no head to re-attach.

Such a story might be possible in parallel dimension theory, on the assumption that the time traveler moves to another universe in his own past, steals the body, and returns to his own universe when he comes forward; but as noted Bender does not travel to the future; he merely awaits it.  Thus he might travel to a universe which is not his past, but when he emerges from the cave, he is in the universe in which he killed Hermes before the accident, and there is no head to re-attach.

Although other theories such as supertime and two-dimensional time come closer to resolving such a story, since Bender waits in the cave history will have caught up with his changes by the time he emerges.

Ultimately the body taken from the past is killed in a freak accident, attributed to the "doom" which attaches to temporal doppelgangers.  Again it seems backwards:  it is not this body but the other whose existence has been undone by temporal interference.  This, though, is necessary to the plot overall, as one of the characters needed to learn that temporal duplicates are doomed.

Next we cut to the chase, as Fry flees a murderous Bender.

After finding the key to time travel tattooed on Fry and sending Bender to rob history of its treasures (and incidentally to encounter himself), the Scammers decide to eliminate the existence of the code by destroying Fry.  Fry uses the code to escape to the past, and Bender is sent to terminate him, and a merry chase ensues.  It is a complicated and convoluted chase, as both parties use the time code again, and then interact with their own duplicates.

Fry goes to half past midnight New Year's Day 2000, moments after he originally fell into the cryo machine that preserved him for the future.  Shortly after Fry's departure, Bender leaves, arriving two minutes before Fry; but that does not mean Bender is already waiting when Fry arrives.  Before Bender can leave from the future, time must resolve between Fry's 2000 arrival and the moment of Bender's departure.  During that time, Fry goes to the pizza parlor, realizes he can't get pizza, returns to the cryo lab where there is cold pizza waiting, and then uses the code to go back an hour to seconds after his original arrival so he can get hot pizza.  This means that before Bender can leave the future, there are two Frys in the past.  They meet; they do not encounter Bender.  The one who has not yet been to the pizza parlor goes there, arriving a minute later, and apparently deciding that a place to live was more important than a slice of pizza--but the other had not already arrived there, because this is a new history.  The double back at the cryo lab remembers that the original has money in his wallet, and in attempting to take it falls into the cryo machine, where he remains until the original Fry gets out and accidently resets the machine for another eight years.

Because the Fry who in our view arrived "second" never made the one hour trip back for hot pizza, the Fry who arrived "first" now cannot exist.  This should create an infinity loop, preventing the future beyond an hour after Fry's arrival, and undoing everything that ever happened in Futurama.  Some would call that a good thing.  However, Futurama has ignored this aspect of its doppelgangers, and so we will follow the thread as if there were a way for a doppelganger to exist by not making the same time trip made by the original.

Before those eight years can elapse, Bender makes most of his trips and begins his terminator hunt.  He arrives two minutes before Fry, but then needs to leave the room unattended for a moment, and so uses the time code to copy himself, meeting the version of himself just arriving from the future.  There are now two Benders; and since they arrived from the future in which there are two Frys, both Frys will arrive momentarily.

(It is at this moment that Tuxedo Bender appears with the tattoo, but he is not relevant to the chase.)

The Fry from a millennium away arrives first, and confronts the Bender who was left on guard; that Bender goes into self-destruct mode and is pushed into a cryo capsule.  The other Fry watches.  Then they talk about the pizza, and the Fry who has not yet been to the pizza parlor goes there and again chooses getting an apartment over getting a slice.

At this point there are two doppelgangers, one of Fry and one of Bender.  One of each pair is in cryo--but it is the opposite one of each.  The "doomed" Bender is in stasis (his doom delayed, although why cryonic suspension would stop a self destruct sequence on a robot is another of those side points that should snap your disbelief suspenders), but the "doomed" Fry is staying in an apartment over a pizza parlor.  Each of those should have caused an infinity loop, but instead Futurama choose to assume that the one who did not travel to the past is doomed, awaiting his fate.

No one likes a cliffhanger, but the chase is the most complicated aspect of the time travel in the movie, and as Bender pursues a doomed duplicate Fry, more nonsense lies ahead next time.

Fry and Bender each duplicated himself, and one of each was put in cryo for the future.  The "doomed" Bender (remember, Futurama doppelganger theory says that a non-time-traveling copy of a time traveler eventually meets with a fatal accident to correct the paradox) already is in self-destruct mode, but the other is pursuing the doomed Fry.

Fortuitously, Bender never paid attention to anything Fry ever said about his life in the past, and so he begins (in another salute to Terminator) with a page from a phone book.

Fry puts the future behind him, trying to forget his love for Leela in the future largely by becoming caretaker for a narwhal named Leelu.  This sends him to the north pole for months when the narwhal is returned to the wild, and brings him back a wiser and prematurely older man.  Finally Bender finds him--and loses him again, and then stumbles on him in the window of the room above the pizza parlor.  He blows up the building, and then joins the untold number of other Benders in the cave to await the future.

What we see in the movie at that point is that Fry survived and saw himself in the mirror, discovering that he now looks and sounds like Lars.  Lars was the assistant at the head museum who stole Leela's heart.  He realizes that all this time he has been in competition with himself, races to the cryo lab, and travels to the future where he anticipates great success winning Leela's heart--having already seen as Fry how that was done.

But it cannot be so; not this time, at least.

Before Fry can see Lars he must be Lars; that means that as Fry sees himself in the mirror, he has never seen the Lars staring back at him.  He needs another reason to make the trip the future, and another reason to take the name Lars and work at the head museum.

We can give him such a reason, though.  Leela rejected Fry because he was so immature.  This Fry is considerably more mature, being and looking older and different enough that he would be unrecognizable.  If he gives himself a new name, he can return to the future and make a more plausible play for Leela's heart.  He has nothing to keep him here--his home was just destroyed, the narwhal he loves left him for another, and odds are good he can't get his old job back delivering pizza.  So he heads for the future, hope in his heart that the girl he has always loved might see something in him if she does not see him as he was.

He does not have that smarmy self-confidence, of course.  Once he injects himself into the future, Fry sees him, and the Fry who escapes from the future becomes the Lars who arrives from the past.  That Lars has every confidence that Leela will fall in love with him, because he saw it happening when he was his own rival.  He goes to play the part and claim the prize, and does so with the confidence of someone who knows the outcome already.  This time through, though, he is hoping, not knowing.

As with all Futurama doppelgangers, Lars is doomed.  Yet it is odd that he lived several years after being duplicated in 2000, survived an attack with a future weapon that destroyed his apartment, then emerging from cryo a thousand years later manages to remain alive long enough for all this to happen around him, and for him to persuade Leela to marry him, and to call off the wedding, and to live through the subsequent distress, before bringing his own death upon himself by pulling the similarly doomed self-destructing Bender from cryo to kill the last of the Scammer aliens.  We might ask how long a temporal duplicate can survive before the code recognizes the paradox, and whether Lars would have lived longer had he married Leela.  After all, everyone is doomed; life has no guarantees.  Lars might have died soon, but so might Fry.  He lasted an inordinately long time for a paradox, and brought his own end on himself.  It is passing odd.

We passed another oddity, a throwaway in the midst of the chase, which we will cover next.

Bender is chasing Fry, but before he believes he has succeeded, he attacks a different Phillip J. Fry.  This one happens to work with Al Gore, who is with him in Florida as the votes are being counted in the 2000 Presidential Election, and from the piles of votes in the box marked "Gore", it appears that Florida will be his--until a stray shot from the attacking robot happens to destroy that box and its contents.

The obvious implication is that Gore would have carried Florida and thus become President of the United States in 2001 had it not been for the completely absurd involvement of a robot from the future.  There are some now cursing the robot and some thanking him.

Setting aside the political statement, it becomes necessary for us to evaluate the impact of this event.  If indeed Gore would have won, then Bush would not have been President in 2001; it is also against all probability that Bush would have been nominated to run against Gore in 2004, and we would never have had the Bush presidency.  Blame the robot, not me.  But that forces us to ask what we would have had instead.

It is difficult to assess what happens with a Gore presidency.  It is rather narrow to think that the identity of the President of the United States would have had much impact on the economic situation.  There is a tendency for Presidents who took the office from the other party (Reagan from Carter, Clinton from Bush, Bush from Clinton) to serve two terms, but coattail Presidents (Johnson from Kennedy, Ford from Nixon, Bush from Reagan) to be ousted after one.  It might be that just as two-term Presidents have low popularity ratings in their sixth year, coattail Presidents are seen as continuing the previous administration and thus of having been in office a considerably longer time.

That means, based solely on the pattern, had Gore won in 2000, he would have been replaced by a Republican in 2004, who probably would have been re-elected in 2008.  Obama would still be an unknown figure, who perhaps put his name into the ring in 2008 but could not overcome the power of incumbency.

Alternately, had Gore (against pattern) served two terms, and had his Vice President Joe Lieberman avoided any pitfalls of politics, it would almost certainly have been Joe Lieberman running on the Democratic ticket in 2008.

There are other possible scenarios, but in nearly every case either Lieberman is the heir apparent to the Gore presidency or the Democrats have failed so drastically that no one they field will stand a chance.  Thus Gore's victory in 2000 would almost certainly prevent Obama's victory in 2008.

Again some are thanking the robot and others cursing him, but we are only concerned with the fact that once the chain of presidential succession has been altered, it is unlikely to get back to whatever it would have been.  Had McGovern beaten Nixon in '72, neither Carter nor Ford would have been in the race in '76.  Had Reagan lost in '80, he would not have been on the ticket in '84.  Without Clinton's shoulders, it is doubtful that Gore would have been the Democratic candidate in 2000.  Thus it is clear that changing the winner in 2000 means it is unlikely that anyone who would have been elected to that office in the future ever will be, or at least, not for the same term.

It is unclear how much impact this has on the future world which is Futurama.  The political, economic, and social impacts of the presidency over the long term could mean a complete change in the human gene pool (because if only one couple does not marry and each marries someone else, over a few generations that will change a very large number of people).  This could change everything about the future, even whether there is a Hubert J. Farnsworth to found the Planet Express Delivery Company.  It may be the single most significant change made by Bender in all of his time travels--and the movie treats it as a throwaway.  But then, that's because it is all a joke, right?  There never really was any chance that Gore might become President, was there?

And now some are cursing me and some thanking me.  Blame the robot.

Once we get past all the other impossibilities, Tuxedo Bender, who began our story by traveling from the end to the beginning to carry the time code back to place on Fry, makes the same trip again.  This time as he waits in the cave, he knows who all those other Benders are, because this time he remembers having been each of them already.  This time, though, he changes history in a drastic way:  he persuades each of his temporal duplicates not to emerge from the cave on schedule, but to wait until the end and emerge together after the Scammers have been defeated, depriving the Scammer aliens of their treasures.

Thus we have Mona Lisa Bender stealing the Mona Lisa, then meeting Tuxedo Bender in the cave, and being persuaded not to exit.  If he does not, though, then Egypt Tomb Bender can never depart to steal the Egyptian relics.  Since he cannot depart, he cannot return, and no other Bender will ever leave to steal artifacts in the past.

It is unclear what will happen from that.  If the Scammer Aliens come looking in the cave for their missing robot, they will find both Mona Lisa Bender and Tuxedo Bender.  The latter is not under their control--they erased the control code.  However, unless the latter erases the control code from the former (and we do not know if he can do that), the former will obey the Scammers and bring the artifact out of the caves.  If asked why he did not come sooner, he will explain that the other Bender told him to wait, and that will lead to trouble.

It seems likely that the Scammers would send Mona Lisa Bender back to destroy Tuxedo Bender.  If he succeeds, Tuxedo Bender is no longer in the cave when future Benders arrive, and does not emerge at the end (nor do all the other Benders), but time is preserved.  If he fails, he will probably be destroyed himself.  That will undo the existence of all other Benders, including Tuxedo Bender, and the code will never be delivered to Fry's tattoo, and we have a complicated infinity loop.

This same problem applies to all other Benders, but it also means that Fry will not flee to the past, Terminator Bender will not pursue him, they will not duplicate themselves, and Lars will never exist.

The film suggests that this is the same temporal duplicate problem it has had all along, and perhaps in a sense it is:  each Bender in the cave represents another that did not make a time trip.  Yet here it puts the doppelganger problem in stark relief.  As we observed early in our analysis, a time traveler who duplicates himself by not making a time trip he already made creates an infinity loop; it is the time traveling version that cannot exist.  Yet here it is difficult to grasp.  The problem is that Mona Lisa Bender never becomes Egypt Bender, yet Egypt Bender already exists.  Thus all the Benders in the cave are duplicates of each other, and under Futurama rules doom will take them all.

--which is what happens as the film is ending and all the duplicate Benders with their art treasures begin meeting their fates.

Hopefully we have now examined all the anomalies in this film, and next time can provide a summary of the disasters as well as the interesting points.

It seems necessary to summarize the entire discussion; it also seems impossible.

It began at the end, since it seems most probable that the Bender who carried the code from the end of the movie to put it where it would be found originally got it elsewhere after the Scammers had been defeated.  His motivations for doing so become irrelevant, since ultimately he takes the code to the past simply because he knows he has already done so.

Our first encounter with time travel as viewers of the film introduces the Futurama approach to temporal duplicates, which involves the paradox-correcting code destroying the wrong copy, leaving the one which paradoxically cannot exist because its history was erased.  This would create a large number of infinity loops, but that the first one prevents the existence of all the others.

When Bender steals the Mona Lisa, he begins a strangely interactive collection of anomalies, since to come forward in time he must hide in a cave with all of his other selves until it is time for him to emerge.  This creates a large number of interlocking sawtooth snaps as his duplicated selves interact with each other, learning about their own futures and modifying what they have not yet done.  Ultimately he changes all of these into infinity loops by converting all the duplicate Benders into paradoxical versions.  And thus as his first trip becomes his last trip, he makes every trip possible and impossible.

The situation with Herme's head needing a body is entirely different, but even more problematic.  No theory of time can explain why the detached head still exists in the future when Bender returns with the body stolen from the past.

The chase is complicated by two infinity loops, as both Fry and Bender duplicate themselves, one of each is cryonically sent to the future, and the others continue their pursuit until Bender believes he has killed Fry, but has actually created Lars.  Lars cannot recognize himself as he does in the film, because he has not yet met himself; but were history able to reach the end he would remember on the replay having seen himself in the future.

Along the way, Bush becomes President instead of Gore, which changes the sequence of Presidential succession, erasing the original history and giving us instead the one we know, including Obama's election, probably altering the future more than any other event in the film.  The Planet Express Delivery Company is unlikely ever to have existed once this change filters through the world.

It is all impossible, at almost every turn.  But then, that's what we expected when we started this analysis.  Futurama never took anything seriously, especially not itself.