FWIW I think the concept of "interesting" is so subjective, time-dependent, and nebulous, that to say there's "only a finite number of interesting" mathematics problems feels absurd to me. What's interesting in ten years time will depend on what other interesting things have been discovered in the intervening time. But you're right, I shouldn't have responded with a cheap joke.

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.