&gt;Cantor&#x27;s diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B.<p>If that were true, why go to all the trouble, just show 1&#x2F;2 which is not a natural number, or sqrt(2) which is not a rational number.<p>Cantor&#x27;s diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up, there will be a number you would miss.<p>The primes and rationals have the same size (cardinality) as the natural numbers, namely countably infinite. See <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Countable_set#Formal_overview_without_details" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Countable_set#Formal_overview_...</a>