Getting a Little Nerdy

Alright today's topic is going to be a little nerdier than zombies. Star Trek. Specifically Star Trek teleporters. Basically what they appear to do is move a person molecule by molecule to a new location. I want to look at this in two different ways. The first, and slightly less problematic, is if we look at this taking place infinitely fast, but not instantaneous. This does mean that it would approximate instantaneous, but would not have some of the same consequences. In this fast version, a person is teleported one molecule at a time. A molecule at point A is destroyed and an exact copy is made at point B. The second way is instantaneous teleportation. That is, all molecules in the body are teleported at once. The person at point A is completely destroyed instantaneously while at point B an exact copy is made.

In the first scenario, we seem to have a more plausible way to maintain identity between the person at A and the person at B. Since molecules are copied one by one there is no mass destruction. The main problem here is at what point does the person at A cease to exist and the person at B begin to exist. We could probably say that A ceases to exist when none of the original molecules are left. But even here we have a problem, because over the course of about 7 years the human body replaces all of it's cells. If we accept that person A ceases to exist when none of the original molecules are left then we must say the person we are today is not of the same identity as the person we were when we were born.

In the second scenario, the person at A is completely destroyed while the person at B is created anew. If someone told you they were going to kill you, but then right after make an atom for atom copy of you would you do it? I personally would not. But if that causes me to lose identity, my current body being completely destroyed and then created anew, what is it that makes up the identity that I have lost? Is it a soul? Is it the body itself? Or something else entirely different?