Faraday probably doesn't exist. It depends on if Ellie left the sunken island.
I'm not sure this is true. Jack, Kate, Ben, etc were on the island in 1977 and did not die. Why would everyone else?
Jack/Kate/etc. time traveled to survive.
Ben was Dharma in '77, and might have been evacuated. Ellie (and Charles) were Others, and the most likely to be on the island when it sunk.But, as of now, we don't know why the circumstances of why the island sunk, short of the assumption that the bomb did it, nor do we even know if Ellie and Charles were Others when the island sunk. It's very possible that, because Jacob died, his "bringing" 815 down wasn't the only event that brought people to the island that didn't happen, like whatever brought Ellie and Charles to the island. But, from what we've seen, I tend to side with Ellie being dead in the sinking, and no Faraday being born.