Toby Bartels wrote:
Prof. Peter Johnstone wrote in part:
The trouble with the Conway construction is not that it's non- constructive, but that it isn't (in any reasonable sense) a construction of the reals. If you stop it at the point when it finally constructs the real numbers 1/3, \sqrt{2}, \pi and so on, then it has also succeeded in constructing lots of non-real numbers like \omega, 1/\omega, 1/2-1/\omega and so on. So how do you distinguish the numbers you want from the ones you don't?
Can you get anywhere by imposing (at that stage) some Archimedean conditions? We need to know first how to interpret natural numbers as Conway numbers, then take the subset of those x such that -n < x < n for some n, and identify x and y if -1 < (x-y)n < 1 for every n.
Pretty much. Peter was speaking of day \omega. With your identification applied to that day, the only other step needed is to remove \omega and -\omega and then you have exactly the field R. A more uniform way of arriving at R is to take the subset consisting of those numbers expressible as a nonempty proper sup or nonempty proper inf in an even number of ways. This is because at day \omega you have \omega, -\omega, D \cdot 3 in the sense of ordinal product, where D is the set of dyadic rationals and 3 = {-1/omega < 0 < 1/omega} (the range of adjustments to each dyadic rational), and the dyadic irrationals. \omega is a proper sup of the integers but not a nonempty inf (though it is the empty inf), and dually for -\omega. The dyadic rationals are not a proper sup or inf of anything (their increments on either side prevent this) but the increment on the left is a proper sup but not a proper inf, and dually on the right. The dyadic irrationals are both a proper sup of what's below them and a proper inf of what's above them. Puzzle. Bearing in mind that on all days prior to day \omega, every number satisfies this criterion, on what other days besides day \omega does this criterion produce a field? Vaughan