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In this paper we introduce a formal model for random access computers and argue that the model is a good one to use in the theory of computational complexity. Results are proved which compare run times for recognizing sets using this model (which has a fixed program) with a stored program model and with Turing machines. The main result, theorem 3, shows the existence of a time complexity hierarchy which is finer than that of any standard abstract computer model. An Algol-like programming language is introduced which facilitates proofs of the theorems.