Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such characteristics are useful for determining when a language cannot be decided by that weak model of computation.
Do Turing-decidable languages have any such necessary characteristics? If not, why not?
Such a characteristic should avoid mention of Turing machines.
Motivation: The only way I know to show that a language is not Turing-decidable is diagonalization, or reducing to a language where diagonalization has already been applied. It would be nice to have another method.
I suspect that there aren't any nice characteristics of this type known, or they'd probably be very well-known; but an argument for why they can't exist could be illuminating.
I'd also be interested in the same question with the phrase "Turing-decidable" replaced with any complexity class, e.g. NP. This question seems more tractable (as there are fewer languages under examination).
Edit: Given the comments, I'll be a bit more precise about what I mean by "characteristics." I would be thrilled with either of the following two possibilities:
1) A property that allowed one to show a language was not in the relevant class (Turing-decidable, NP, whatever) without using diagonalization.
2) A property that can be stated in a language too weak to define Turing machines.