[...] it has always been a little pet peeve of mine that this idea about aliens is called a paradox. Let's see here, on the one side we have the claim that aliens exist. On the other side it is pointed out that we have yet to find them. How are these two ideas contradictory? I could claim that there is probably some life in the ocean we have yet to discover. If you claim that he have yet to find them, that does not mean we have hit a contradiction of sorts. It only means that I am either wrong about my first assertion, or that we simply have not found said life yet. The resolution is simple. A paradox should not have such an easy and apparent resolution if it has one at all.
The Fermi paradox:
The Milky Way galaxy is so vast and old that, according to all we know about physics, chemistry and probability (see the Drake equation), life should have arisen in many different places.
If aliens exist, why are they not here?
The speed of light limit is NOT the answer to this question - you see, an intelligent species can colonize the entire Milky Way in a few tens of millions of years and the galaxy is BILLIONS of years old.
If aliens don't exist, then why don't they exist? What's the reason for life's scarcity?
And why is there life on Earth? Are we the winners of a lottery with practically non-existent chances of success?
These unanswered questions (no one was able to offer a reasonable answer to them) are what make the Fermi paradox a paradox.