That actually has an interesting failure. The first interesting number that's interesting because it's not special works. The second might qualify, the 10008 does not.
Because I constructed a set of otherwise boring numbers. The first otherwise boring number can arguably be interesting, but if you accept numbers are not inherently interesting by being on the number line you need find a reason to remove them from the otherwise boring list.
I think the word “otherwise” is key here, and makes the situation completely different. The paradox arises from a self-referential definition, and that word removes the self-reference.
Zeno’s Paradox is solved with real numbers and calculus. This paradox is explicitly on the natural numbers, meaning you have a well defined notion of “smallest” for any nonempty set of them.
I don’t think the solution is related to time. Rather, it’s simply a matter of recognizing that it’s ok for something to be well defined that does not exist. The notion of an interesting natural number, such that it’s a binary property and being the smallest natural number, just doesn’t work. Thus we can conclude that there’s no such thing as “interesting” as defined like this. You can fix it by making interestingness a continuous value, or using a definition such that being the smallest number in a set isn’t enough to be interesting.
There are a bunch of ways to solve Zeno's Paradox involving time (and not necessarily involving real numbers and calculus), Wikipedia gives a quick overview of many.
In the interesting number paradox a solution can be related to time (or recursion depth) such that, as the person you were responding to mentioned, the first few otherwise not interesting numbers can be considered interesting without considering all not otherwise listed numbers as interesting. The wikipedia page gives a different example solution involving time.
