There are some theorems that go along the lines of "all reasonable properties of <math subject> are computationally undecidable." Here are two examples:
- Rice's Theorem: "all reasonable semantic properties of programs are undecidable."
- Adian–Rabin Theorem: "all reasonable properties of finitely presented groups are undecidable."
Even though I only know of two examples, I suspect that there are many theorems like these. Is there a list of these kinds of theorems?