&gt; You can prove that if you try to map each integer to some rational number, there will be some rational numbers that are not on that list --- there are more of them.<p>Did you mean reals here?  There _is_ a (bijection) mapping between integers and rationals.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Cantor_pairing_function#Cantor_pairing_function" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Cantor_pairing_function#Cantor...</a>