Hypercomputation refers to computation surpassing the Turing model, not just exceeding the von Neumann architecture. Algebraic constructions yield a ÿnitely 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

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... es 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 deÿnition.