And when you meet people, and someone says to you " Hey, Libman, you still owe me that five bucks", then together with the first thing, and given your brain is maintaining all the chemical balance and everything, there's no way of escaping being yourself. The episode of Northern Exposure comes to mind, where they put Chris Stevens to trial and his line of defense was that he no longer is the same person who commited the fellony years ago, I don't remember right now but I guess in the end he looses the case with this. So perhaps the socially derived criterion may be even more important ( and I know that hurts an Objectivist ).
So if your identical clone showed up tomorrow and the rest of society recognized him as you, then you wouldn't object to your own death just because of that clone?
That is a form of mortality we might not be able to escape, as our minds and bodies gradually experience the
Trigger's Broom paradox - you will survive as a copy of a copy of a copy... You may be immortal in the perception of others, and perhaps in the memories of your new copies (in as far as memory can be authenticated, which isn't by much), but your old copies will die.
C'est la vive.
I don't want this pessimist perspective to discourage anybody from their interest in longevity / immortality, however, because it ought to at least be possible to extend life experience as we know it today far into the second and possibly even third centennial without the need for any major brain modifications, and a lot of new ideas regarding preserving "stream of consciousness" identity will probably be developed in the meantime.
BTW, since you brought up political philosophy first, I'd like to add that this is also an excellent argument for inheritance rights / against the death tax (not the worst of all taxes, but still theft). People inheriting wealth might seem like an exception to the capitalist argument that in a free society wealth reflects merit, but the inherited wealth reflects the merit of the person who created it, and it is his Right to spend it however he sees fit, including giving it to somebody else, which might have been his incentive to creating this wealth in the first place. Keeping your own property as parts of your mind and body are gradually regenerated is not much different - if Bill Gates lives to be 10,000 he probably won't be the same person who've founded Microsoft, but his Right to his bank account will remain legitimate.