So, you are using numbers to count and characterize other groups of numbers (sizes and cardinalities). Got to be suspicious about the conclusions drawn. It would seem to be like using the lens of a telescope to examine the optics of another. Invisible distortions and characteristics of the former would be confused with characteristics of the observed latter telescope. Are we raising ourselves by our own bootstraps? A mathematician thinks logically, but I am not formally trained. So, tell me how I am wrong.