{A,B,C} is of greater cardinality than {A,B}.  However, the argument that &quot;a rule works for finite numbers, so it must work for infinite numbers&quot; is clearly false.<p>For a your mapping, see:  <a href="http:&#x2F;&#x2F;math.stackexchange.com&#x2F;questions&#x2F;512397&#x2F;is-there-a-simple-constructive-1-1-mapping-between-the-reals-and-the-irrationa" rel="nofollow">http:&#x2F;&#x2F;math.stackexchange.com&#x2F;questions&#x2F;512397&#x2F;is-there-a-si...</a>