It starts with a simple concept, in each generation the number of your ancestors doubles. You have 2 parents, 4 grandparents, 8 great-grandparents and etc. Some of my family lines go back 18 generations. At that level, I would have over 262,000 direct ancestors. By 20 generations the number would increase to 1,048,576. Continuing backward, I would ultimately have more ancestors than the total number of people who have lived on the earth. How does this mathematical concept correspond to reality? At some level, it would appear that everyone in the entire world is somehow related. This is called the Ancestor Paradox.
However, in these ancient times, it is supposed that there were far fewer people than at the present time. (By the way, if you ever want to see every type of way-out theories from UFOs to asteroid impact, try looking up ancient population figures). So how do the calculated figures of my ancestry correspond to the supposition that there were fewer people living in the past? See Wikipedia. Where are all these missing ancestors?
The easy answer is to acknowledge that some of your ancestors married relatives. Would the impact of intermarriage reduce the actual number of ancestors enough to explain the missing grandparents? It would if everyone married their brother or sister, then there would be no increase in the number of ancestors. But, I have yet to meet anyone who can document an ancestral line of only marriages between brothers and sisters. Another alternative would be to marry only first cousins, then the numbers would increase only by two in each generation instead of doubling. Following this line of reasoning soon gets you to another paradox called Zeno's paradox. I'll let you look that one up.
The seemingly simple concept of the doubling of ancestors in each generation quickly becomes a morass of math and supposition. There are three very citing articles on the subject by Brian Pears beginning in 1985 in the Northumberland & Durham Family History Society Journal. The three articles are: Our Ancestors, Conceptions, Misconceptions and a Paradox, The Ancestor Paradox Revisited and the Ancestor Paradox Yet Again. Let's just say that I don't find all of the logic in Pears' articles convincing. Before you get going on this question, you can begin by noting the fact that the Ancestor Paradox is not a Heap Paradox question. Neither is it a Bald Man Paradox because those paradoxes rely on the problem of fuzzy sets. (You can look those up also).
In some ways the problem is related to the ancient Theseus's ship paradox. (I know, you can look it up). The real question is whether or not the Ancestor Paradox is a problem in logic or math or is really a issue that can be solved. As was stated by Giangiacomo Gerla in his treatise called Why I have an extra-terrestrial ancestor, "The paradox lies in the possibility of confuting a theory on factual reality by a purely logical argumentation.
From a genealogist's point of view, every ancestor of a given person can be identified, just as we accept the proposition that every living human being had a mother and a father. However, we also jointly assume the opposite of this fact to be true, that it is practically impossible to identify every ancestor of any given person. So it may be logical from examining the simple mathematical progression, to assume that the number of ancestors increases in every generation, when in fact that progression may not be true. The number of ancestors would only increase if each ancestor is considered to be unique. So the problem is not so much one of a paradox but a fallacy. Although I said I did not agree entirely with Pears, I do agree that the flaw in the exposition of the Ancestor Paradox lies in a fallacious assumption, that is, the uniqueness of our ancestral lines.
Any thoughts on this subject? Comments would be appreciated.