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Not really, no. Generational ships might make colonizing the nearest star systems possible, but even colonizing our own galaxy would require some kind of suspended animation. The milky way is between 100,000-200,000 light years in diameter so even at the speed of light, you're looking at a travel time that is ~33-66% of the time that humanity has even existed(homo sapiens are currently estimated to have become a distinct species 200,000-300,000 years ago)... just to go to ONE star system out of the hundreds of BILLIONS that exist in our galaxy. You're gonna need generational ships so self-sustaining and capable that the generation that actually arrives at the destination will have long forgotten the point of the trip and might not want to leave the comfort of the ship.
Still, colonizing our own galaxy is at least theoretically possible, given enough time. The real filter is just how unimaginably large the universe is. The vast, VAST majority of the observable universe is FOREVER out of our reach, as it is expanding away from us faster than the speed of light. Then there's the unobservable universe, which could literally be infinitely bigger than the observable universe for all we actually know.
That's why faster than light travel is the holy grail. Without it, we're just kind of stuck.
Imagine if wormholes had zero constraints on the physical location of the other side of the wormhole though. We could open a portal to OUTSIDE the observable universe. What a mindfuck. We might even find a false vacuum decay racing towards us at the speed light, or regions of space that are contracting instead of expanding, or initiate a new big bang by opening a wormhole to an area of space where that hasn't happened...we could travel to a point where we can watch the milky way get formed, since the light of its formation is just reaching that region of space. If it turns out the heat death of the universe is just a local phenomenon, we could continue expanding forever beyond it. World without end.
Generational ships wouldn’t have to reach the edge of the galaxy, just the next planetary system. There’s no reason civilization needs to remain centered on Earth, either. Think of it as a wave traveling outward, where it eventually reaches the edge, by many smaller hops. It will also eventually reach earth, where they might wonder at signs of a prehistoric civilization. Actually, think of it like the Middle East, where empires rise and fall, crusades and jihads burst through, religions rise out of nowhere, people speak many different languages. A galactic civilization could be dynamic and ever changing, distance can make us strangers to each other, the fate of any planet matter only to its inhabitants and neighbors
Sure, that's an option. It doesn't really change my overall point though that anything beyond galactic colonization is unrealistic on any time scale. Our next nearest neighbor, the Andromeda Galaxy, is over 2.5 MILLION light years away, over 10 times farther than my "crossing the milky way" example, with nothing in-between to make a pit stop if needed, you have to cross the true void of space to get there.
And that's just to get the next nearest galaxy. Current estimates suggest the observable universe contains 2 TRILLION galaxies.
True, you’re not getting to the next galaxy. However within the galaxy, your generation ships only need to work for a century or two per voyage. That’s at least conceivable