I guess the next step of the argument would go something like "we can formalize certain forms of interesting* knowledge in the relationships among prime numbers

I guess the next step of the argument would go something like "we can formalize certain forms of interesting* knowledge in the relationships among prime numbers, which are infinite" (e.g. Godel's work). After which point we can get into a very long conversation about what constitutes "interesting" and what constitutes "knowledge" . It would certainly be a feat to formalize any of this, and we'd still be debating definitions and assumptions if we could formally prove anything. > Learning the next thousand thousand prime numbers is not very interesting if you already know the first few million, and of very limited practical use. Maybe your intuition here is right but my hunch is that if you were able to prove this conjecture, it would absolutely blow the socks off the mathematics community. I look at it as a Pascal's Wager w/ regards to knowledge: either there's more to know that we don't know we don't know, or there isn't. If there is, we'll likely only find it out if we believe there is more knowledge to uncover and spend our time trying to uncover it. My overall hunch here is that a lot of our current knowledge is constrained by our mental semantic structure, which is bounded by the working set of objects we can mentally manipulate at an instant in time. That set is probably O(10) for most of the population, and likely O(100) for everybody. We can increase the number of semantic objects we can operate on by working on them temporally using logical rules, but those rules themselves are bound by this O(100) limit, and so on for any meta-rules. I would find it very surprising if all interesting things that could be stated were relationships between O(100) semantic objects, however abstract.