&gt; <i>Where am I going wrong?</i><p>In short, infinities are complicated and intuition doesn&#x27;t work well.<p>In slightly longer, you&#x27;re talking about the difference of the rate of growth of two functions rather than actually about the cardinalities of the sets.<p>We define two sets as having the same cardinality when we can create a bijection between them.  We can list the primes in order from smallest to largest and number them with the natural numbers.  So we&#x27;ll have 2 match up with 0, 3 with 1, 5 with 2, 7 with 3, etc.  Every single prime number will correspond with a natural number AND every single natural number will correspond with a prime number, no exceptions.  So they must be the same size.  They are also the same size as the integers and the rational numbers but the set of real numbers is a bigger infinity.