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If we think in terms of computation, an infinite sequence can also be modelled by a computer program, churning out number after number onto a long tape (or these days your hard drive). At any given point in time, there are only finite many outputs. As long as you keep supplying more tape, electricity, and occasionally additional memory banks, the process continues. The sequence is not to be identified by the `completed output tape’, which is a figment of our imagination, but rather by the computer program that generates it, which is concrete and completely specifiable. However here we come to the same essential difficulty with infinite processes: the program that generates a given infinite sequence is never unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with `infinity’. It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same.
Unfortunately there is no such theory.
With sets the dichotomy between finite and `infinite’ is much more severe than for sequences, because we do not allow a steady exhibition of the elements through time. It is impossible to exhibit all of the elements of an `infinite set’ at once, so the notion is an ideal one that more properly belongs to philosophy—it can only be approximated within mathematics. The notion of a `completed infinite set’ is contrary to classical thinking; since we can’t actually collect together more than a finite number of elements as a completed totality, why pretend that we can? It is the same reason that `God’ or `the hereafter’ are not generally recognized as proper scientific entities. Both infinite sets, God and the hereafter may very well exist in our universe, but this is a philosophical or religious inquiry, not a mathematical or scientific one.
