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Let me venture a possible application, since there are already many responses celebrating pure mathematics. As far as we understand, the natural numbers are no

Let me venture a possible application, since there are already many responses celebrating pure mathematics. As far as we understand, the natural numbers are not sufficient for modeling physical phenomena. The reals/complex while immensely useful also occasionally turn out to be “too complicated” to give theoretical guarantees/proofs of models working well. On the practical side, that means that these models/algorithms can’t be guaranteed to not give junk results, while working on the domain of real numbers. Now, purely speculatively, since the naturals/rationals are aleph0 in size and the reals are aleph2, that means there likely exists a set of numbers of size aleph1 sitting in between the two. What if we could use that set of numbers to construct our physical models? Could we somehow guarantee better behavior… Eg: in quantum mechanics, or field theory, or chaos, etc?! :-)

drdeca 5y ago
Whether the cardinality of the reals is aleph 1, aleph 2, or something larger, is independent of ZFC. In a model of ZFC in which the set of reals has cardinality greater than aleph_1 , I wouldn’t be surprised if there is a subfield of the reals of cardinality aleph_1 , but I would be surprised if such a field was useful for things like that. Such a field would, of course, not be complete with respect to the usual metric on the rationals, so we wouldn’t have the desired convergence properties. We wouldn’t really be able to do infinite sums in it? (Well, perhaps some other sense of infinite summation could be done, but it wouldn’t be the usual sense.) In addition, because such a subfield would only exist as an uncountable proper subfield in some models of ZFC, I find it hard to imagine that it would allow computations that wouldn’t otherwise work? I suppose it could motivate some computations which would then also work regardless of what model of ZFC is being used / is true ?

drdeca 5y ago

Whether the cardinality of the reals is aleph 1, aleph 2, or something larger, is independent of ZFC. In a model of ZFC in which the set of reals has cardinality greater than aleph_1 , I wouldn’t be surprised if there is a subfield of the reals of cardinality aleph_1 , but I would be surprised if such a field was useful for things like that. Such a field would, of course, not be complete with respect to the usual metric on the rationals, so we wouldn’t have the desired convergence properties. We wouldn’t really be able to do infinite sums in it? (Well, perhaps some other sense of infinite summation could be done, but it wouldn’t be the usual sense.) In addition, because such a subfield would only exist as an uncountable proper subfield in some models of ZFC, I find it hard to imagine that it would allow computations that wouldn’t otherwise work? I suppose it could motivate some computations which would then also work regardless of what model of ZFC is being used / is true ?