Don't tie yourself up in knots worrying about infinite sized objects is my advice!
It really is pretty meaningless, but since you ask if the span is infinite the wing area is also infinite so CL can be 1/infinity, so your alpha is just 1/infinity more than the zero lift alpha, however much lift you need. If the increase in alpha is 1/infinity then the incremements in zero lift drag are also infinitessimal.
Intuition is not a handy tool for analysing infinity.