Mike Barr writes:
Perhaps Vaughan is unfamiliar with the following definition of the reals. I got it from Steve Schanuel. I believe that he got it from Serge Lang and he probably got it from Emil Artin.
I believe the idea *came* from Artin but I don't believe Artin was thinking of it as a *construction* of the reals. I was so taken by the construction that I thought it should be better known by tertiary teachers. I wrote a little note An efficient construction of the real numbers, Gazette Australian Math. Soc. 12 (1985) 57-58 on the construction (giving credit to Schanuel and reporting that Peter Johnstone told me Richard Lewis from Sussex had also come up with it). [Warning and Challenge: Steve pointed out that my note messes up the construction of the sup operation.] Happy 2000 plus. Ross