Well, yah, an infinite number of them exist. I'm curious, though, how many of these principles a la the Halting Problem actually hold if they are excluded?
We take as doctrine that the halting problem is generally undecidable, but I haven't heard anything about any results for the Halting Problem (minus self referential paradoxes). Which is concerning because in practice nobody gives a crap about self referential paradoxes and they very rarely actually arise.
There is Rice's theorem (which is simple extension of Halting Problem and a basic theorem in computability theory), which states that all non-trivial, purely semantic properties of programs are undecidable. For more details, see https://en.wikipedia.org/wiki/Rice%27s_theorem .
The Wikipedia article gives 2 proofs for Rice's theorem.
a) As a corollary of Kleene's recursion theorem. This proof seems to rely on quines, and therefore relies on the same principle of self-referential paradox?
b) As a proof by reduction to the halting problem.
I'm not very confident in my interpretation of (a) ...
We take as doctrine that the halting problem is generally undecidable, but I haven't heard anything about any results for the Halting Problem (minus self referential paradoxes). Which is concerning because in practice nobody gives a crap about self referential paradoxes and they very rarely actually arise.