The explanation of Cohen's filter for making new reals reminds me of surreal numbers {L | R} but in this case every real is in either L or R. Still, I fail to s

The explanation of Cohen's filter for making new reals reminds me of surreal numbers {L | R} but in this case every real is in either L or R. Still, I fail to see how that gives you more than restricting L and R to contain only rational numbers...