For what it's worth, Olber's paradox uses the wrong formula for the volume of a shell some distance R from earth... The formul as I remember it from Olber's paradox is 4*Pi*R^2*dr, where dr is the thickness of the shell. However, this value only approaches accuracy as R approaches infinity. It is wrong for all finite values of R.
And I was not double counting anything. The actual volume of such a shell is: 4/3*Pi*(r+dr)^3-4/3*Pi*r^3. This is a value that is admittedly less than proportional to R^3, but more than proportional to R^2 for any finite value of R greater than zero. This volume is actually even greater than the value that Olber was utilizing, and dividing it by R^2 to calculate the expected intensity of radiation in that entire shell that reaches a point at distance R does not approach 0 as the distance approaches infinity.
But the real problem with Olber's paradox is not the miscalculation of the volume of the shell at some distance from earth,and in turn the number of elements within that volume which will emit radiation,but rather with the assumption that the universe is somehow actually infinite in the first place.
Olber's paradox revealed that trying to make an assumption that the universe might be infinite is flawed, and by my understanding helped to serve as an impetus at the term to find alternative explanations for what we observed, eventually leading to the widely accepted big bang hypothesis.
And observed red shift means that objects are moving away from each other, which means that at some point they were much closer together, and rewinding the clock even further suggests that the universe began at a single point, and has been expanding outward ever since (although you can no more find a point in space that is the center of it than you can find the center of an inflated balloon anywhere on the surface of the balloon).
Bottom line: the universe is finite. Even if it were ever found to be expanding into an infinite unbounded space.