It goes like this
1. Earth has intelligent life on it.
2. Even if intelligent life is extremely rare, the galaxy has hundreds of billions of stars, so there should be other intelligent life elsewhere. Even one other planet with interstellar travel technology is enough.
3. Even at slow speeds, an technological civilization can visit every star in the galaxy in less 50 million years. (The 50 million is actually a bit of an *over*estimate)
4. The galaxy is old enough that there were planets as old as Earth is now at the time the Earth first formed.
5. That’s plenty of time for a technological civilization to get to the solar system and leave evidence of that.
6. There is absolutely no evidence that this has ever happened, so where the hell are they?
There are plenty of answers to the paradox, but they generally fall into assuming that aliens choose to not come to the solar system. Remember that they can visit literally every star in the galaxy, so them not coming here makes us a special case that needs explanation. There’s an idea called the Copernican principle that we should assume we’re average without evidence otherwise.
Alternately, there could be no other life in the galaxy which is odd for two reasons. The first is that life isn’t made out of anything special. You’re pretty much made of methane, ammonia, water, and carbon dioxide linked together in complicated ways, and the ancient Earth was covered in those chemicals. Life also appears at pretty much the earliest time it could, so it seems reasonable to assume that any planet like the early Earth will end up with life like the Earth.
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