  Arrow's Impossibility Theorem establishes bounds on what we can require from
voting systems. Given satisfaction of a small collection of "fairness" axioms,
it shows votes can only exist as dictatorships in which one voter determines
all outcomes. Votes are modelled as maps from a collection of partial orders,
the preferences of voters, to a single verdict which is another aggregated
partial ordering. This result is classic and has an extension called the
Possibility Theorem that shows these dictatorships needn't exist with infinite
voter sets. Mihara extends this work by examining the computability of each of
these results. He found that the only voting systems that are in any sense
computable are necessarily dictatorial, which takes away from the usefulness of
the Possibility Theorem.
  In this paper we primarily survey the results of Mihara, focusing not on
applied consequences, as much of the surrounding literature does, but going
into greater details on the underlying Mathematics and Computability of the
proofs. We give detailed exposition on the methods used and introduce all
notation. We first see complete proofs of the classical results and a
sufficient introduction to computability that an unfamiliar reader should be
able to follow without prior knowledge of the field. We then expand into
Mihara's results, and using our established knowledge of computability show the
problems with trying to compute non-dictatorial social welfare functions. This
involves introducing an extended definition of computability called pairwise
computability.
