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

Question:

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

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?

Best Answer - Chosen by Asker: 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.