It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.
Edit: actually, forget about the above. I just find it very annoying when people dismiss good conversations with not-so-good jokes.
I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.
Secondly, I stand by the statement that there is no merit to this joke. This is because the way it defines interesting is very hand-wavy. There are interesting and non-interesting problems, but by a sleigh of hand you can turn the non-interesting problems interesting, thus proving that basically everything in the universe is interesting. At least in the mathematically describable universe. When everything is interesting, nothing is interesting. So we can dismiss the proof as a silly joke.
What mathematicians find interesting is a different story. However, we can almost certainly say there is only a finite number of problems mathematicians as physical beings can solve. If we have 200 mathematical symbols at our disposal, and we consider all strings of these symbols of length 1000,000, we have captured all the descriptions of problems that fit to 1M symbols. But that's a finite number. Going beyond that starts to be difficult for a human to grasp (if 1M is not too much already), so all mathematical problems that are solvable by a physical mathematician are in that set of strings. And that's not even saying anything about whether or not they're interesting..
By the way, if I'm not completely mistaken, Gödel's argument to show the incompleteness of mathematics relies on encoding all mathematical statements as numbers. So I'm certainly not being very original here.
Godel's incompleteness theorem technically relates to individual axiomatic theories (i.e. the set of facts that logically derive from a given set of axioms). The numerical encoding you refer to applies to logical statements within that theory. Arguably, the kind of mathematics that humans do is not constrained to a single theory, but is a more general form of reasoning that is often flexible about which axioms may or may not be assumed.
This isn’t an enormously important point - the actual question at issue is an empirical one, “in a steady state, can we produce interesting problems at a rate that exceeds our ability to solve them and integrate our understanding” or something like that - but I did rankle at a “trivial” proof which is invalid due to equivocating between multiple definitions of the word “interesting” (which should really take an object, “interesting to me” vs “interesting to something smarter than me”).
I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.
I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.