I'm going to start by stating the obvious: you should only read this if you've already finished BioShock Infinite, or if you don't mind having the plot hung in front of you in its birthday suit. If you want to experience it first, then bookmark this shizzle and come back when you have.
You're back? Good.
Another thing: if you haven't finished it, then why aren't you playing it right now? Go on, play it. Would you kindly go play it? Do you even row, bro? Do you?
Anyway, as the title helpfully suggests, there may be massive spoilers after this line break.
in·fi·nite: Limitless or endless in space, extent, or size; impossible to measure or calculate: "an infinite number of stars". The definition of the game's namesake, and the core story concept, revealing its true importance towards the end of the story.
During the conclusion of BioShock Infinite, Elizabeth defends against the Songbird by teleporting the trio to Rapture. She performs the inter-dimensional equivalent of a brain fart, taking them to a place she has seen through one of her tears, in a bid to save them from the relentless metal pigeon.
The songbird dies, trapped in the sea, in the place where I think it may have been born.
The maternal instinct that the machine had for Elizabeth, clearly has some similarities to Rapture's Big Daddies. Comstock may have seen Rapture through a tear and reached in, stealing some of its secrets. I also don't think it's a coincidence that Columbia's Vigors bear many similarities to Rapture's Plasmids. With this in mind, the songbird dying in Rapture has a strange poetry to it.
They leave, shortly after, in a Bathysphere, emerging at the base of the first game's lighthouse. Surrounded by thousands of stars, it thrusts out of the ocean, poking through a clear, obsidian sky, untouched by light pollution. Elizabeth looks up at the stars in wonder, and says, "look at that - thousands of doors, all opening at once", whilst gazing up at their beauty.
The stars are revealed to be lighthouses, all of which are interconnected by walkways, each a door to a different universe. This is the 'many-worlds interpretation' of quantum theory - an infinite number of universes, where every possible outcome has occurred. Each world the same, yet different. Assuming you played the game, your experience wouldn't have been identical to mine. We wouldn't have jumped the same amount of times and fired the exact same amount of ammunition, so in essence, each playthrough of BioShock Infinite is another universe. No two Bookers take the exact same journey, yet regardless of choice, they reach the same conclusion. Much like the choice between a bird and a cage, the journey is irrelevant, it's the destination that matters.
To give you some examples: in one universe you may have taken your car to work, yet in another you may have walked to work and ended up under a car. In another you might have had diarrhoea and never made it to work that day at all, yet in a different one you didn't even have a job, but you smelled bad. All choices, all consequences, both big and small, as infinite as space itself. Worlds of constants and variables.
In BioShock's universe the constants are that there's always a lighthouse, a man and a city. It happens because it does, because it has, and because it will. It's like Schrödinger's cat: alive, dead and a mixture of the two states. Thinking about it too much makes your head feel like exploding, and actually, in one universe it most likely has.
In the game's universe(s?), all of the choices and combinations of events lead to the same place, with Booker trying to save his daughter from himself. This reality is true for both Booker and his baptised alter ego, Comstock. Both are opposite versions of the same person, with the turning point being whether Booker accepted a baptism in an attempt to repent after the sins he committed at Wounded Knee - and giving his daughter away.
Comstock labels Booker as the False Shepherd, who will lead his heir - the Lamb of Columbia - astray. He uses religious propaganda to fuel Columbia's citizens' belief that the one who bears the mark 'AD' on their right hand is the false shepherd. In truth, Booker branded himself with the initials of his daughter (Anna DeWitt) to remind him of his wrongdoing.
The baptism takes place twenty years prior to the start of the game, which is just after Booker gave up baby Anna (Elizabeth), wiping away the debt he owed, which we'll get to later.
In Columbia, it becomes apparent that Comstock had been trying to produce an heir, yet he had been made infertile by exposure to 'tears' created by the Lutece twins' machine. A cruel irony, as Comstock was funding the research as a way of looking for the daughter that he had abandoned in a previous life.
An unbaptised Booker gives his baby to Robert Lutece, who tells him that the debt is paid, and Mr. Comstock washes him of all his sins.
The debt, if you were wondering, is Booker's heavy conscience. The way the debt was to be wiped, was by Robert Lutece taking him to a new universe, and as a result, his memories would be wiped. This set of events goes around and around continuously, with the cycle ending up as constant in itself. Booker, Comstock and Elizabeth are stuck in a temporal loop, of sorts.
If Comstock had never looked for his daughter, then baby Anna never would have been taken from Booker, thus never having to look for her in the future, and possibly lessening his need for redemption. But this is a world of constants and variables, and in the case of the search for - and abandonment of - Elizabeth, it can be categorized into the former.
So let's just assume that Booker is fated to chase his daughter, who he gave up to himself, for eternity. What is the point of his journey? And how did he get there? After all, the universe that contains Columbia can't be his own. How can it, when Comstock is Columbia's founder, and this Booker is from a universe where he refused the baptism and never became Comstock?
How then did Booker end up in this universe, and why? The answer lies with the Lutece twins. As I mentioned earlier, it was all part of the deal.
The Lutece 'twins' aren't actually twins, you see. Rosalind Lutece and Robert Lutece are actually gender-swapped versions of the same person. They brought the Booker that we control to this universe in a bid to conduct a practical test of a 'thought experiment', promising him that he would be wiped of all his bad memories.
Presumably their intentions were to break the circle of events, which would essentially prevent their deaths at the hands of Comstock. "Will The Circle Be Unbroken", being the song Booker hears when entering the church in the entrance to Columbia.
On the boat at the beginning of the game, Rosalind tells Robert that she didn't believe the thought experiment would work practically, and that they shouldn't be conducting an experiment that has already failed. This tells us that they have done this before. The dead body in the lighthouse teasing the possibility that it could be another Booker who has failed.
The fact that the twins have done this before is confirmed when you meet them again. They ask Booker to flip a coin. The chalkboard originally shows that the coin has landed 'heads' twelve times, and once Robert turns around he reveals 110 more chalk marks, signifying that this is the 123rd Booker DeWitt that they've used.
Robert's argument in the thought experiment leans towards the fact that if an infinite number of universes exist, then there must be one where their manipulation succeeds.
So why does Comstock use the knowledge of the twins to capture his daughter? He sees a vision of the future through a tear, he sees his heir, Elizabeth, setting the sodom below afire with his wrath. His religious beliefs and his self-proclaimed holiness have clouded his humanity, and he dreams of revenge on a country that had forgotten his sacrifices. All of his citizens see him as a prophet because he has seen the future, and the power has driven him mad.
Comstock - although he and Booker are essentially the same - is completely different to Mr. DeWitt.
Even on an aesthetic level they're different, with Comstock looking obviously older. Comstock may actually be older than Booker, but travel through the multiverse may have drained his youth, just as it may have also tainted his mind.
He traps Elizabeth in a tower, both to protect her from Booker, and to prepare her for succeeding him when Booker inevitably kills him. She is locked inside, with books about alternate dimensional travel and lockpicking to keep her company. All the while, Zachary Comstock drains her power, in a bid to keep her contained.
Elizabeth is powerful and Comstock knows this, which is why he keeps siphoning her. We know that Comstock had to use machinery to snatch Booker's baby, so how does she have these powers naturally?
My theory, is that she got her powers when Comstock ordered the portal to be closed prematurely, causing the loss of Anna's finger. Now I'm not proposing that she has a 'magic finger', but this event caused her to physically exist in two separate universes. Maybe this imbued her with the abilities she has, and this may also explain how the Lutece twins flit between worlds with such ease.
So, her powers are unleashed when the siphon is destroyed, and she sees all of the outcomes and possibilities at once. She can see all of the doors. She knows the only way to break the chain, and takes Booker to where it all began - the baptism.
Booker agrees to murder Comstock in his crib, and Elizabeth takes him to where the monster was born so that he may achieve this. She reveals the truth to Booker, who accepts this is how it must be. She drowns him. This causes Comstock to never have existed in the first place.
But how did killing a Booker from after the baptism stop Comstock from existing? My theory is that this Booker had just had his debt wiped away when we take control of him at the start. The Lutece twins take him to the lighthouse soon after he gives up Anna, wiping his memory in the progress.
Once Booker had crossed over, however, his mind got confused and he started to manufacture new memories from his old ones. "Bring us the girl, and wipe away the debt". Every time his memories are repressed or he tries to remember a repressed memory his nose bleeds, signifying the effect that inter-dimensional travel has on the subject.
So Booker dies, and Comstock never was. We then see an alternate Booker, back in his apartment. He runs to the crib, but the camera cuts before we see if the baby is there or not. That part is open to interpretation and there is no way of knowing whether she was there or not. With there being an infinite number of universes though, we can rest assured that Booker and Anna are out there, living a normal life, somewhere in the stars.