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. --Toby