Your original question was "So you don't think that humans are finitely realizable physical systems?" responding "No" to that question does not entail that MN thinks that humans are not finitely realizable physical systems - it's a subtlty concerned with the scope of negation which may have escaped you. He may believe, for instance, that the notion of a finitely realizable physical system, or indeed even the notion of a human being, is not clear enough to be able to reach any reasonable conclusion concerning whether one is an instance of the other or not, and in which case the reasonable position is probably to suspend judgement. — jkg20
The realization Rosen talks about is not a system that operates on symbols but is a "mapping" like you take 10 baskets, put a few apples into each and then have realized the first 10 members of it. Using the laws of mathematics to manipulate symbol-systems is very different from this. A computer printing the infinity-symbol on a sheet of paper is not a realization of infinity.The most natural approach to take seems to be the following: let f:A —> *B be an arbitrary mapping. If f is to be physically realizable, it is no restriction to take A and B to be countable sets.
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