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) ...
Options include not letting this bother you (my favourite) and just ignoring it and hoping it doesn't come up, as indeed, it often doesn't.