  For over a decade, the hypercomputation movement has produced computational
models that in theory solve the algorithmically unsolvable, but they are not
physically realizable according to currently accepted physical theories. While
opponents to the hypercomputation movement provide arguments against the
physical realizability of specific models in order to demonstrate this, these
arguments lack the generality to be a satisfactory justification against the
construction of \emph{any} information-processing machine that computes beyond
the universal Turing machine. To this end, I present a more mathematically
concrete challenge to hypercomputability, and will show that one is immediately
led into physical impossibilities, thereby demonstrating the infeasibility of
hypercomputers more generally. This gives impetus to propose and justify a more
plausible starting point for an extension to the classical paradigm that is
physically possible, at least in principle. Instead of attempting to rely on
infinities such as idealized limits of infinite time or numerical precision, or
some other physically unattainable source, one should focus on extending the
classical paradigm to better encapsulate modern computational problems that are
not well-expressed/modeled by the closed-system paradigm of the Turing machine.
I present the first steps toward this goal by considering contemporary
computational problems dealing with intractability and issues surrounding
cyber-physical systems, and argue that a reasonable extension to the classical
paradigm should focus on these issues in order to be practically viable.
