Hypercomputation by definition
Hypercomputation refers to computation surpassing the Turing model, not just exceeding the von Neumann architecture. Algebraic constructions yield a finitely based pseudorecursive equational theory (Internat. J. Algebra Comput. 6 (1996) 457-510). It is not recursive, although for each given number n, its equations in n variables form a recursive set. Hypercomputation is therefore required for an algorithmic answer to the membership problem of such a theory. Yet Alfred Tarski declared these theories to be decidable. The dilemma of a decidable but not recursive set presents an impasse to standard computability theory. One way to break the impasse is to predicate that the theory is computable--in other words, hypercomputation by definition.