Here is a puzzle I thought of the other day.
Suppose a time traveler from the future proposes a deal for you. He has taken some scans of you and he claims to be able to reconstruct you precisely in the future - molecule for molecule. Furthermore, in the future improved advances in medicine have made eternal youth possible, so that what you get is not just more life but eternal life. In return, he asks you to run errands for him for a mere decade or so.
Should you accept? Our immediate intuition is to say no - whatever he will reconstruct in the future will not be you - perhaps it will be a copy of you, but not you, who will have died by then.
Consider, though, a different scenario. Suppose a professor of biology from university X claims to have found the key to eternal youth. He claims that the cell-division process is inherently flawed; every time your cells divide, the new copies are slightly degraded. Worse, some cells in your body do not divide at all, and as a result degenerate over time. He claims to have perfected the cell division process, and learned how to induce division in cells which do not divide.
He offers you a pill - for the same price as the time traveler in our first example - which, if you take, will induce each of your cells to make a perfect copy of itself every day - no degradation. If you take it, he says, you will have eternal life. Some of your friends have taken it and have indeed stopped aging as a result. Moreover, they seem the same as ever - no side effects.
Should you accept? Our immediate intuition is to say yes - this is eternal life, and surely its worth whatever is asked for it.
But the two scenarios are actually identical - in the latter case, you will be copied one cell at a time, but in the former case you will be copied in one shot. Accepting that no cell of your body now is going to be part of the future you, it seems weird to argue that whether the future entity is you depends on how it was constructed from you - whether on a cell by cell basis, or all-at-once.