[Edited, because it was wrong.]

The doomsday argument is,

O(X) = random human me observes some condition already satisfied for X humans

pt(X) = P(there will be X humans total over the course of time)

pt(2X | O(X|2)) / pt(2X) < pt(X | O(X/2)) / pt(X)

This is true if your observation O(X) is, "X people lived before I was born", or, "There are X other people alive in my lifetime".

But if your observation O(X) is "I am the Xth human", then you get

pt(2X | O(X|2)) / pt(2X) = pt(X | O(X/2)) / pt(X)

and the Doomsday argument fails.

So which definition of O(X) is the right observation to use?