Hypercomputation

Hypercomputation is the theory of methods for the computation of non-recurisve functions. The classes of functions which they can compute is studied in the field known as recursion theory.

Hypercomputation was first introduced by Alan Turing in his 1939 paper Systems of logic based on ordinals, which investigated mathematical systems in which an oracle was available to compute a single arbitrary (non-recurisve) function from naturals to naturals.

Other posited kinds of hypercomputer include:

A quantum mechanical system which somehow uses (for example) an infinite superposition of states to compute a non-recursive function.
A Turing machine which is running for an infinite period of time (perhaps the observer is being dropped into a black hole).
A Turing machine which is accelerating exponentially (in a Newtonian universe, such a gadget might operate by manufacturing a clone of itself which was only half the size and operated at twice the speed).
A nondeterministic Turing machine which has a preference ordering over its final states.

At this stage, non of these devices seem physically plausible, and so hypercomputers are likely to remain a mathematical fiction.

References

Alan Turing, Systems of logic based on ordinals, Proc. London math. soc. 45, 1939