Let's take a simple look at the method of mapping counting numbers to rationals:
You can't do it with a finite limit N, and its gets WORSE as N gets bigger. Then suddenly, and magically, infinity makes it all work itself out. NO! With convergent series things getter better as N gets bigger. With divergent series things get worse. This is a divergent problem handled badly.

Another Cantor related subject is the Cantor set, generated by taking the number line between 0 and 1 and removing the open inner third part an infinite number of times. The question is then asked as to what, if anything, is left: http://lesliegreen.byethost3.com/articles/counting.pdf

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