Why? Because feeling the pain of a chemical burn is better than feeling nothing at all.
Tyler Durden isn't a separate person. Tyler is the Narrator’s ID—the raw, unfiltered masculinity, rage, and creativity that the "IKEA man" suppressed. When the Narrator shoots a bullet through his own cheek to kill Tyler, he isn't just killing a villain. He is killing a part of himself.
He lets Tyler cut his lip. He lets Tyler pour lye on his hand. He lets Tyler sleep with the woman he loves.
We don’t even know his name.
And that is exactly the point.
This is where the character becomes fascinating. He isn't a hero. He isn't even particularly brave. He is a man who is so sick of his own passivity that he invites Tyler Durden—chaos incarnate—to move into a dilapidated house on Paper Street.
We can call him “Jack.” We can call him “The Narrator.” We can call him “The Space Monkey.” But the brilliance of Chuck Palahniuk’s Fight Club (and David Fincher’s film adaptation) is that the main character is a deliberate void. He is an empty IKEA catalog floating through a sterile life.
His tragedy isn't that he is poor or oppressed in the traditional sense. His tragedy is that he has confused having with being . He doesn't want love or meaning; he wants the right coffee table. He defines his soul by the catalog of items he owns.