I thought about this question for a longer time. There is a standard proof by contradiction that there is no smallest positive rational/real number $r$ by considering $r/2$.
Now what irritates me about that proof is that it relies on the assumption that such an $r$ or the algorithm for constructing such an $r$ could be given explicitly if it existed. Maybe it just can not but it still can exist. Maybe it can only be addressed by introducing a new way for the description of infinitely small numbers or, equivalently, the numbers that "lie between arbitrary large numbers and infinity". If we do not have a way of talking about such a regime of numbers, then how can we comprehend a set like the natural numbers as it is one object with infinitely many numbered elements but not containing infinity? Normally, infinity is not included in the natural numbers because one would argue that if it was a number, then one could find the number just before infinity but that would not be possible. But what if we introduced a way of talking about such a number? A way of talking about a smooth transition from infinity to numbers that we can write down? Maybe the name or algorithm for the smallest positive decimal number would just take an infinite amount of time to be written down. What if there are more numbers than there are names?