It's true that a family tree either dies out or grows exponentially, but 2 is not necessarily the relevant exponent.
If the expected number of children an average descendant of Slytherin has is N, after 33 generations we should expect to see N^33 heirs of Slytherin. (You should think that this manipulation is suspicious as well, but it can be justified mathematically in, say, the Galton-Watson model.) Taking N=1.25 gives us only around 1500 descendants after 1000 years. And this is, if anything, an overestimate that does not take any intermarriage into account.
Powers of 2 only become relevant if you're looking backwards from a specific person; for example, if you want to know whether two people have a common ancestor. In that respect, I (tentatively) believe the paper you link to.
One of us must be wrong; it can't both be the case that everyone 1000 years ago is an ancestor of everyone living today, and that the average person 1000 years ago only had 1500 descendants.
I think N is closer to two or higher; assuming the average person has two children, they will have four grandchildren on average, eight great-grandchildren on average, and so on. So there really should be 2^33 heirs, though not 2^33 unique heirs; many of those heirs are just different genealogical paths to the same people.
I think if N were below two, it would be below the replacement rate and the population would shrink over time.
Two new short chapters! Since the next one is coming tomorrow and we know it'll be short, let's use one thread for both.
This is a new thread to discuss Eliezer Yudkowsky’s Harry Potter and the Methods of Rationality and anything related to it. This thread is intended for discussing chapter 105 (and chapter 106, once it comes out tomorrow). EDIT: based on Alsadius' comment about thread creation for MOR chapters, let's also use this thread for chapter 107 (and future chapters until this nears 500 comments) unless someone objects to doing so. Given that this is the final arc we're talking about, thread titles should be updated to indicate chapters covered.
There is a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author’s Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: