I believe that pedigree collapse is not the correct - or at least not the only - answer to the problem or paradox of the ever growing number of ancestors.

The number of ancestors does *always* grow, unless it is forced to decrease because of lack of partners.

Even under modest assumptions about the collapse rate the number of ancestors will grow *exponentially* - only with a smaller base than 2.^{1}

So in any case the number of ancestors of an individual will eventually reach the number of all living humans at a past point in time - sooner or later.

Pedigree collapse does only delay this point in time. So you nevertheless face the problem of "as many ancestors as living humans".

Once you accepted that, it's quite easy to see a solution of the paradox: Beyond your "point of equality" (between ancestors and living humans) *all* humans have to be your ancestors.

So the question only is:

Under which assumptions can one's point of equality be calculated?

Another one:

Is there still a recursion law like (*) beyond the point of equality?
How does it look like?

^{1}
Consider the maybe too simplified recursion law for the number a(n) of ancestors in generation n+1:

a(n+1) = 2·a(n) - p·2·a(n) = 2·(1-p)·a(n) (*)

This means that each of one's ancestors in generation n has two parents diminuished by a certain percentage p of those parents that happen to be the same person. (*This* is in essence pedigree collapse.) It gives - for larger n - the number of ancestors

a(n) = (2·(1-p))^{n}

which still is an exponential law.

With a constant collapse rate of p = 0.25 - which means that cousin marriage is the rule - the number of ancestors grows like 1.5^{n} (compared to 2^{n}) which still implies 10 millions ancestors after 40 generations.

Note, that for p = 0.5 (sibling marriage) there is no growth at all!

Not a stupid question at all. It would be amusing to ask it on various forums (I came from math.stackexchange.com) and see the different types of response you get. – Ross Millikan – 2013-02-04T05:14:25.780