What is wrong with the maths? Or the geneology? Or the logic?

Question

Each of us has two parents [well apart from Immaculate Conceptions of which only one is historically recorded]

and each of us has four grandparents for the same reason.

So for each generation we go back we double the number of ancestors at that generation?

GEN1(1944) 1: me, you, anybody
GEN2(1909) 2. parents
GEN3(1876) 4. grandparents
GEN4(1842) 8. great-grandparents
GEN5(1813) 16. great-great-
GEN6 32.
GEN7 64.
GEN8 128.
See where this is going?
GEN16. 32,728
My ancestors trace back 30 generations in Ireland and we can track about 5 more back through Wales, England to Normandy around the time of the Norman Invasion of England in 1066.
GEN32. is actually 2 to the power of 31 ancestors
which considerably more than the whole population (at that time) of the British Isles and France.

Have I that many ancestors?
Has biology failed me?
Or Maths?
Or logic?

Answer 1

At some point there was some intermingling of your ancestors.
Example: your parents could share the same great-great-great grandparents. This would reduce your calculations. Consider that this may occur numerous times.

Answer 2

Imagine an island with 100 people; 50 men, 50 women. They all marry and have just 2 kids, one boy, one girl. The kids mature and marry each other. They again have two kids. When the grandchildren turn 10 or so, the original 100 people die. The generation 3 kids marry each other - they marry their first or second cousins. This pattern keeps up for 15 generations. The island never has a population of more than 300 (100 soon-to-die grandparents, 100 parents, 100 children under age 10). There should have been 2 to the 15th original people; there were not.

That is an oversimplification. Most of us have ancestors in recent times - after 1800 - who married a cousin. In the middle ages, when most people lived their whole lives within 10 miles of the hut they were born in, it was common.

Answer 3

Your logic has failed, but only a bit. Remember than your parents had you, but they also had your brothers and sisters. Also, many of us had ancestors who were distantly related to each other - very distant cousins. After only a few generations, the genes are not closely related enough to create physical problems, for the offspring, and indeed, after a few generations, most people have no idea who all their cousins are!