The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers.<p>Cantor&#x27;s diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B.<p>Just because you can map two infinity&#x27;s to each other does not mean they are of the same size consider: Limit(0-&gt;inifinity) of (x - (x&#x2F;2)) algebraically that&#x27;s clearly Limit(0-&gt;inifinity) of X&#x2F;2 which is infinity.<p>PS: What makes Cantor&#x27;s diagonalization interesting is you can repeat it recursively an infinite number of times.  This is more obvious in base 2.