Using negative results to prove positive results in computability theory Many results in cryptography depend on impossibility results/conjectures in complexity theory. For example, public-key cryptography using RSA is believed to be possible because of the conjecture about infeasibility of the factoring (and the modular root finding problems).

My question is : 


  do we have similar results in computability theory? Are there interesting positive constructions using negative impossibility results? 


E.g., does the undecidability of the halting problem allow us to perform tasks that we would not be able to do if the halting problem was decidable?
In some sense, this is what the theory of parametricity is all about. 

Data abstraction is how we ensure that no client of a module can access the elements of a module except according to the interface exposed by the module. 
We rely on this to ensure that the internal invariants of data structures cannot be broken by the clients of a module -- eg, if you access a balanced tree only via the published interface, then it follows that the tree will always be balanced. 

So we use a negative property -- that no possible client can break the abstraction boundary -- to deduce a positive one -- that the data representation invariant always holds.