I am assuming that Jin's grave and headstone are just a memorial correct? I doubt that his body was ever recovered from the blown up freighter and flown back to Korea.

Well since the "outside world" thinks he died in the airplane crash then no his body is not there and


Jin is alive


If you look at the head stone it says that the date of death is the day the plane crashed.

If Sun told everyone that Jin died on a frieghter it would blow everyone elses cover story apart. To her she saw Jin die on the frighter. So to her he is dead. Now you can't go back to Korea and not have a funeral if she knew Jin was alive as again this would debunk the cover story. Hence she must have a funeral with or without a body, since he died out in open sea it's unlikely you're going to have much of a corpse on an exploding ship. If you check the date of death on the headstone it states the original date when the plane crashed in Sept 2004. So her cover story is that Jin died when the plane crashed. But since in her mind Jin died the pain was real for her.