Imagine the following scenario: Astronomers have found an asteroid on a collision course with earth. It will destroy out civilization and kill all humans on the planet. However, that won't happen for another ten years. A large number of the world's top scientists have come together to try and solve the problem, and they have concluded that we will not be able to deflect the object from hitting Earth. The only chance for the continued survival of humanity is to send a spacecraft full of people, which might survive in space for the several thousand years necessary for the earth to become habitable. Each person in the world is given a choice: If you dedicate the next ten years of your life to helping build the space station, you will be entered into the lottery for a ticket onto the space station. You can give the ticket to whomever you want. Otherwise, you can just relax for ten years, and join the biggest party of mass hedonism since Caligula. What would you choose?
So the scenario doesn't say what your chances of getting a ticket are. Most likely, that would be determined by how much work the people get done in ten years. How many people are necessary to repopulate the world? Preliminary research (i.e. google search) points to somewhere around 1,000, to avoid inbreeding. So probably the best strategy is to have as many self-contained satellite habitats orbiting the earth, each with 1,000 people. And then hopefully at least one of these will survive. Another strategy is to have just one massive space ship, but that's a bad idea since then it only takes one big mistake or accident (or crazy person) to destroy it.
So I'm looking at the situation where for two to three years, the scientists perfect a few models of space ship, then every factory in the world starts making as many of them as they can. Wikipedia says that in the last 50 years or so, only about 550 people have been launched into space. But our technology has grown exponentially (even though we are no longer capable of going to the moon) If we can devote a significant fraction of the world's GDP to this project, then we might be able to produce a few dozen of these space stations. . If we devote all our time and money to this problem, maybe we will be able to launch a few dozen of them before time runs out. So that's about 12,000 people who will be saved. At the same time, I would estimate that about half of the world's population would support this effort, that's about 3,000,000. That's a 1/25,000 chance for each person. A tiny chance, but perhaps worth it, if you know that you did your part in helping humanity avoid extinction?
Over several thousand years, each space station will undoubtedly develop its own culture and civilization, even if there is constant communication between space ships. Will they attack each other? If our space station's air filter breaks, will we raid another's, dooming them and saving ourselves? More likely, all space stations will work together, helping each other
Next is the problem of recolonization. When all the surviving space ships land, there might be some hostility between them. One issue is when, exactly, to land. If our space ship lands earlier than the others, we will have more time to set up a prosperous city and multiply. Later than the others, and we will have a more hospitable Earth to greet us. However, most likely there will be more than enough Earth for everybody, so this hostility probably won't cause human extinction.
A different possibility is to make a space craft that is capable of building more copies of itself from raw materials (moon rocks? asteroids?). That way we just launch a few "seed ships" and in a few millennia there will be hundreds or thousands ready to land. And in that case some may not want to land, but instead might move to other parts of the solar system (or to different stars!). This actually seems to me like a necessary step in the development of our civilization. With space stations that are not only self-reliant but also are capable of reproducing, we change all of space into a possible habitat for humanity. And in that case there's no need ever to return to Earth. We might just send it a post-card, though.
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