@TECHREPORT{Wischik97aformalisation, author = {Lucian Wischik}, title = {A formalisation of non-finite computation}, institution = {}, year = {1997} }

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Abstract

Abstract. Abstract Recent work in the field of relativitistic space-times suggests that it may be possible for a machine to perform an infinite number of operations in a finite time. Investigation of these machines has three motivations. First, because the machines may be physically possible, they have implications for the general question of what problems are theoretically solvable. Second, the physical laws that govern their operation give rise to a rich mathematical structure. Third, by investigating general relativistic computation, we get a clearer picture of properties peculiar to the special case of Turing computation. A mathematical formalisation of the operation of the machines is presented, and shown to correspond to their physical operation. It is proved that there is no satisfactory way to give a finite description for the machines. Although Gödel sentences exist if attention is restricted to a finite set of machines, further results [Ear94,Hog96] about the computational power of the machines and their equivalence to the Kleene arithmetical hierarchy are shown to depend upon arbitrary assumptions. This gives rise to a non-finite version of the Church-Turing thesis. The case of non-finite computation is used to arrive at an abstract principle of computation that is independent of physics. A dissertation submitted to the University of Cambridge