If the duplicate says "I did X on my nth birthday" it's not true since it didn't even exist.
Correct.
So what does the duplicate say when I point out that it didn't exist at that time?
When talking to you, or someone who shares your attitude, my duplicate probably says something like "You're right, of course. I'm in the habit of talking about my original's experiences as though they're mine, because I experience them as though they were, and both I and my original are perfectly happy talking that way and will probably keep doing so. But technically speaking you're quite correct... I didn't actually do X on my 9th birthday, nor did I have a 9th birthday to do anything on in the first place. Thanks for pointing that out."
Which is closest to your last option, I suppose.
Incidentally, my duplicate likely does this in roughly the same tone of voice that an adoptive child might say analogous things when someone corrects their reference to "my parents" by claiming that no, their parents didn't do any of that, their adoptive parents did. If you were to infer a certain hostility from that tone, you would not be incorrect.
It gets very difficult to call this "memory."
It's not difficult for me to call this a memory at all... it's the original's memory, which has been copied to and is being experienced by the duplicate. But if you'd rather come up with some special word for that to avoid confusion with a memory experienced by the same body that formed it in the first place, that's OK with me too. (I choose not to refer to it as "knowledge of what the original did", both because that's unwieldy and because it ignores the experiential nature of memory,, which I value.)
but then in what sense is it a duplicate?
Sufficient similarity to the original. Which is what we typically mean when we say that X is a duplicate of Y.
"I'm in the habit of talking about my original's experiences as though they're mine, because I experience them as though they were" appears to be a form of delusion to me. If somebody went around pretending to be Napoleon (answering to the name Napoleon, talking about having done the things Napoleon did, etc) and answered all questions as if they were Napoleon but, when challenged, reassured you that of course they're not Napoleon, they just have the habit of talking as if they are Napoleon because they experience life as Napoleon would, would yo...
In June 2012, Robin Hanson wrote a post promoting plastination as a superior to cryopreservation as an approach to preserving people for later uploading. His post included a paragraph which said:
This left me with the impression that the chances of the average cryopreserved person today of being later revived aren't great, even when you conditionalize on no existential catastrophe. More recently, I did a systematic read-through of the sequences for the first time (about a month 1/2 ago), and Eliezer's post You Only Live Twice convinced me to finally sign up for cryonics for three reasons:
I don't find that terribly encouraging. So now I'm back to being pessimistic about current cryopreservation techniques (though I'm still signing up for cryonics because the cost is low enough even given my current estimate of my chances). But I'd very much be curious to know if anyone knows what, say, Nick Bostrom or Anders Sandberg think about the issue. Anyone?
Edit: I'm aware of estimates given by LessWrong folks in the census of the chances of revival, but I don't know how much of that is people taking things like existential risk into account. There are lots of different ways you could arrive at a ~10% chance of revival overall:
is one way. But:
is a very similar conclusion from very different premises. Gwern has more on this sort of reasoning in Plastination versus cryonics, but I don't know who most of the people he links to are so I'm not sure whether to trust them. He does link to a breakdown of probabilities by Robin, but I don't fully understand the way Robin is breaking the issue down.