The Ending of Bioshock Infinite Explained

In case you are too dumb to take the hint from the title:

SPOILERS!!!!

are after the jump:

So I assume everyone left here has finished Bioshock Infinite, correct?  To start off, let's summarize the very basics about what happens after you destroy the Siphon:

1. Elizabeth taps into her full power, allowing her to see and travel to an inconceivable number of parallel realities (possibly infinite?) and time travel.
2. She takes you to Rapture, killing Songbird (note: this proves she can travel through time, as Rapture is bother ~60 years after Columbia, and presumably in a different reality).  You take the bathesphere up to the surface and enter a lighthouse.
3. Entering the lighthouse reveals other versions of you and Elizabeth, walking around an apparently infinite number of lighthouses.  Elizabeth explains that each lighthouse represents a parallel reality.
4. It's revealed that OH SHIT BOOKER IS COMSTOCK! (in different realities, depending on if he chose to be baptized)
5. It's revealed that OH SHIT ELIZABETH IS ANNA, aka A. D., aka. BOOKER'S DAUGHTER!
6. It's revealed that OH SHIT BOOKER GAVE ANNA AWAY TO THE LUTECES AND BY EXTENSION COMSTOCK!
7. It's revealed that OH SHIT ELIZABETH HAS HER POWERS BECAUSE BOOKER GOT HER PINKIE CUT OFF IN A TRANS-REALITY TEAR!
8. Consumed by regret, Booker spends the next twenty years wasting away - until the Luteces offer him a chance to get Anna back.  He steps through a tear into - the beginning of the game!  His brain constructs new memories to reconcile with the parallel reality, explaining why he didn't know any of this going in.
9. Elizabeth decides the only way to stop Comstock is to "cut him off at the root".  Booker volunteers to be drowned by Elizabeth(s) at the baptism.  When he drowns, all the parallel Elizabeths but "our" Elizabeth blink out of existence.
10. After the credits, Booker wakes up back in his office, and hears a baby crying in Anna's room.

Whoa!  That was quite a roller-coaster of an infodump.  While watching it, it's easy to get caught up in the emotion (which resonated pretty well).  However, if you're anything like me, you probably rolled the thing around in your mind - and read things on the internet like this - to see if it was internally consistent and held up to scrutiny.  Many gamers and fans of other media share this compulsion to make sense of the universes in which we invest.  

Many people seem to think that the ending doesn't really hold water, but I've done a lot of reading and thinking about it, and I think I've got it worked out.  

The first thing to understand is that all realities share "variables" and "constants".  This is directly stated by Elizabeth and evidenced by the actions of the Luteces.  For instance, when they have you randomly flip a coin early in the game, the board clearly indicates that you have flipped heads every time.  That's a constant across all realities: in that situation, Booker will always flip heads.  On the other hand, when they have you choose a choker for Elizabeth - either a bird or a cage - they make it clear that you don't always pick the same one every time.  That's a variable: a choice that could go different directions, and the very core of what makes parallel alternate realities a thing.  Basically, think of realities as a tree: every  variable creates new branches - new parallel realities.

We also know that the events of Bioshock Infinite have happened many times before, but the one you play through seems to be the first time that you and Elizabeth succeed in erasing Comstock.  The Luteces have clearly been running different versions of Booker through Columbia and even experimenting on him, as the coin flip board and many other clues throughout the game make clear.  It's highly likely that we are playing the 123rd iteration of Booker to attempt to rescue Elizabeth, as the number 122 appears too many times throughout the game to be a coincidence.  However, in the end it doesn't matter which iteration we are.

110 + 12 = MIND BLOWN

[side note: there is a theory that every time you die in the game, you are actually replaced with another version of Booker from another reality, and what you see of Elizabeth reviving you is actually memories constructed by your brain to cope with a new reality.  It doesn't really matter to our discussion if this is true, it only matters that we keep in mind that multiple Bookers have already tried and failed]

Now that we've established the basics, let's follow the timeline of relevant events that lead up to and include Bioshock Infinite.  

1. Booker participates in the massacre of innocent non-combatants at Wounded Knee, and is subsequently consumed by guilt. 
2. To deal with this guilt, Booker considers religion.  At the baptism, he either commits to religion and is baptized, or refuses the baptism. 

From this point forward, we'll refer to the version that is baptized as Comstock, and the one that isn't baptized as Booker.  Keep in mind that only one can exist in each universe, as the splitting of universes occurs because of his choice at the variable point.  First, let's follow the Comstock timeline.

1. Booker accepts the baptism.
2. Booker embraces being born again.  He changes his name to Comstock.  He feels that the baptism washed his guilt away, and he embraces the idea that he did nothing wrong.   This explains how someone who started out as Booker could become so racist and jingoistic: in order to believe that what he did at Wounded Knee was actually right, he'd have to believe that other races are subhuman (so slaughtering them was A-O.K.) and he'd have to believe in the righteousness of the American exceptionalism and expansionism that led to Wounded Knee.  This explains the twisted ideology of Columbia coming from a man who in other realities doesn't hold those beliefs.
3. In both realities, Lutece discovers the "Lutece Field", which is a way to suspend an atom in the air using quantum mechanics.  The female Lutece in Comstock's reality and the male Lutece in Booker's reality discover this simultaneously, and subsequently discover the existence of parallel realities by using the Lutece Field to communicate via Morse Code between realities. 
4. Comstock meets Lutece and convinces Congress to give him funding to build Columbia, using the Lutece Field to levitate the buildings.  Lutece also builds a machine that creates "tears" between realities, and Comstock frequently looks through them to get his "visions of the future" (actually just looks at alternate realities) to build up his following and reputation as a "prophet". 
5. As Columbia's construction continues, the Fink brothers use the tears to steal music and scientific breakthroughs from other realities, leading to anachronistic songs on Columbia as well as the creation and sale of vigors and human/machine hybrids like Songbird and the Handyman.  It is very possible, though not confirmed, that the Finks got that technology by looking into the reality of Rapture and the original Bioshock, explaining the similarities between vigors/handymen and plasmids/big daddies. 
6. Comstock finds that his prolonged exposure to the Lutece Machine has accelerated his aging and made him sterile, but he must have an heir.  He uses the tears to find a reality in which he had a child (Booker's reality) and gets Lutece to develop a way to travel between the realities. 
7. Comstock steals Anna (and the male Lutece from Booker's reality comes along into Comstock's reality), renames her Elizabeth, and presents her to the public as a miracle child that grew in his wife's womb in just 7 days.  
8. Mrs. Comstock threatens to go public with Elizabeth's origin, so Comstock kills her and blames one of his house servants, Daisy Fitzroy. 
9. Elizabeth begins to develop her powers, so the Luteces create the Siphon, which steals most of her power to fuel the city. 
10. Columbia is launched.  It eventually intervenes in the Boxer Rebellion against U.S. government wishes, then secedes from the Union and disappears into the clouds.
11. The Luteces begin to have doubts about Comstock and what they are doing to Elizabeth.  Comstock has Jeremiah Fink sabotage the machine that makes the tears in an effort to kill them.  Instead, the machine's malfunction effectively knocks them loose from time and space.  They are now essentially untethered, and can exist in any reality at any point.  The extent of their mobility and power is unclear. 

This is where Comstock's reality is left when Bioshock Infinite begins, so before we proceed, let's follow Booker's much less eventful timeline:

1. Booker refuses the baptism.
2. Booker meets his wife and they have a baby together, Anna.  His wife dies in childbirth. 
3. Booker is still consumed by guilt over Wounded Knee and basically gives up on life after his wife dies.  He descends into gambling, financing it as an amoral hired gun breaking up strikes for Pinkerton.
4. Booker is approached by his reality's Lutece, who offers to "wipe away the debt" (I am still not clear if the "debt" refers to some unspecified gambling debt or to the debt Booker feels to the world for his participation in Wounded Knee) in exchange for his daughter. 
5. Booker hands Anna over to Lutece (this is a constant, which is why Elizabeth says "you don't leave this room until you [give her to him]")
6. Booker has second thoughts and chases Lutece into an alley, where he finds Comstock holding Anna and about to go through a tear back to his own reality.  Comstock and male Lutece step through the tear, but Booker manages to get a hold of Anna.  He and Comstock struggle, and as the tear closes it cuts off the end of Anna's pinky, with the rest of her body crossing over with Comstock.  
7. Booker spends the next twenty years in a deep depression. 

And that's where Booker is when Infinite begins.  

The Luteces are unstuck from reality, and formulate a plan to repair their predicament, partly fueled by the male Lutece's insistence that they fix the mess they made by helping Comstock steal the girl.  The Luteces open a tear to Booker after he has spent 20 years mourning his mistake of giving up Anna, and offer him a chance at redemption.  Booker comes through the tear into Comstock's reality, and his mind constructs new memories to reconcile with that new reality, explaining why he doesn't remember the previous events and only has a vague directive to "bring back the girl".

Over the course of the game, there are many choices the player makes, some of them constants and some of them variables.  However, if he survives, they generally lead to the same outcome.  Booker winning the lottery at the beginning of the game is a constant, but who he throws the baseball at is variable (though both choices lead down the same path).  Other variables may or may not make a difference, but in the end they don't really matter in the grand scheme of things, because the very act of finally succeeding leads to the events of Bioshock Infinite never having happened.  The only variable we really care about is the baptism.

In order to get back to reality, the Luteces have to eliminate every single reality that would result in them becoming unstuck.  Even if they eliminated some of those realities, as long as just one remains that knocks them loose from time and space, they remain loose in all realities by the very nature of having been knocked loose.  Tracing the event that knocked them loose back through the constants that led up to it, they must eliminate all realities in which Comstock kidnapped Anna, and that only happens in realities that are paired with another reality containing a version of our Booker.  Therefore, for our purposes of parsing the ending, we are only concerned with those universes.  We can infer that in all other universes, the Luteces never become unstuck from reality.

So they try to create circumstances that would eliminate all of those realities by going to these "paired" realities and bringing Booker over to attempt to free Elizabeth, eventually leading Elizabeth and Booker to go back to the baptism and prevent Comstock from ever coming into existence.  The Luteces do this by trial and error, conducting experiments, such as the coin flip, along the way.  When one Booker fails, they jump to the next pair of universes and try again, intervening in new ways to try to create the desired outcome.  It takes them 123 tries to create a set of circumstances in which Booker chooses to be drowned at the baptism, thus snuffing out every reality in which Comstock would have come into being.

Whew!  So now that we have all the background we can talk about why the ending works, despite there appearing to be logical inconsistencies.  Let's look at those things that appear to be inconsistent and explain them in light of the facts we've already established:

This would make for a great porno if they weren't all your daughter...

1) Why are there multiple Elizabeths in the final scene?

Recall that while travelling amongst the lighthouses, we see multiple other versions of Booker and Elizabeth also doing so, but we do not see one for every lighthouse.  From this we can infer that some, but not all, of the previous* Bookers succeeded in unshackling Elizabeth.  But if they succeeded, how come subsequent attempts were needed?

Because Booker's choice to die to prevent Comstock from coming into being is a variable, and the previous "successful" Bookers refused.  The Luteces had to go back to the drawing board until they created the exact set of circumstances that would eventually lead us, the 123rd Booker, to choose drowning.  We can only speculate at this, but I suspect that having Booker initially fail and see the version of the future in which Elizabeth destroys New York City may have been a factor.  It's likely the previous "successful" Bookers actually defeated or evaded Songbird and freed Elizabeth without experiencing that reality, and therefore did not fully grasp the importance of stopping Comstock**.  This elegantly explains why the "correct" path for the Luteces to achieve their goal still included an apparent failure on Booker's part.  Their previous attempt to lead Booker to more directly "win" may have led him to free Elizabeth and go to the lighthouses, but still did not result in the action (drowning) that would free the Luteces.

Great, now we're going to have to go all the way to another reality for decent pizza!  Thanks, Grandma!

The other Elizabeths in the final scene are those that previous Bookers successfully freed, but "their" Bookers did not choose to die to prevent Comstock from coming into being.  Since all of the other "successful" runs of the Luteces' experiment happened at the same time (but in different realities), those other Elizabeths are arriving at that moment at the same time - but only "our" Elizabeth has a willing Booker in tow.

2) Why the hell would killing Booker at the baptism just the one time get rid of all Comstocks?

Recall what I said before about the parallel realities being like a tree, with branches sprouting at each variable.  Well, that means that every single branch of reality that results in the Luteces becoming unstuck from reality comes directly from the baptism.  In the metaphor of the tree, all realities we care about spring from that one event.  That means that all realities we care about spring from one reality.  Sure, there are certainly an infinite number of parellel realities already going on by the time the world gets to that baptism.  Millions of years of variables saw to that.  But there is only one reality in which circumstances were such that Booker makes the baptism decision that eventually leads to the Luteces' predicament.  Perhaps in other realities, Booker dies at Wounded Knee, or is never born?  Since we have already established that the Luteces only need to eliminate all realities in which they become unstuck, we can safely infer that none of those other infinite other parallel realities that don't include the baptism decision lead to them becoming unstuck.  

Therefore, the moment just before Booker decides whether or not to be baptized is one singular reality, and none of its infinite parallel realities matter for our story.  It is truly the single trunk of the tree from which all of the problematic branches spring.  That's why the Luteces only need Elizabeth to intervene at that one moment in time in just that one version of reality.  It is the last point at which there is only one active branch that leads to them becoming unstuck.

3) In the after-credits scene it is implied that Booker is alive, has been returned to 20 years ago, and he has baby Anna back!  How is this possible!?

Oh, dear reader, I'm glad you stuck it out with me this long.  This is the best part.

Booker could not have drowned if he didn't accept the baptism.

Think about that for a second.  We have all these paired realities in which one Booker took the baptism and one didn't.  Then we travel back in time and drown Booker as he's being baptized.

Which means none of the versions of him that refused the baptism are drowned!

I know it's a little bit of a brain bend, but it's true.  Only the Bookers that would eventually become Comstock died.  All of the futures in which Booker would survive the baptism and become Comstock were snuffed out - but only those realities were snuffed out.

So now every version of Booker that refused the baptism is still alive, still has Anna, and has no "paired" Comstock to come interfere by taking his baby.  Therefore, he was returned to the last moment before Comstock interfered: the day Lutece came to offer to take the baby (which he no longer will).  

So elegant!

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I hope that the preceding wall of text has helped you understand the ending of Bioshock Infinite a little better.    I know that, like any work of art (yeah, I said it!), there are many interpretations, and I don't mean to say that my way of thinking about it is the only valid way.  I tried to back this up with as much evidence as available, but it does require a bit of speculation and inference that is not explicit in the game itself.  If you have a different way of looking at the ending, please share it in the comments!

*Though I use the term "previous", that is only true from the Luteces' point of view.  From our point of view, all of the other 122 attempts happened at the same time, just in parallel realities.

**It's interesting to speculate on the consequences of allowing Comstock to succeed.  Though we witness the destruction of NYC, we are told that the plan goes well beyond that: to invading and conquering all other universes.  It very well may be that one of the factors motivating the male Lutece to put things right is that his actions lead directly to many or all universes eventually being invaded or destroyed!