> In article <NvudnSgQHKv-VyrZnZ2dnUVZ_tSdnZ2d@speakeasy.net>, > Hatto von Aquitanien <abbot@AugiaDives.hre> wrote: > > [snip] > >> With a well ordered finite set, we can count from either end. Reversing >> the order makes sense, etc. We can't do that with infinite sets. > > Not so; it can easily be done for SOME infinite sets. For example, the > set of integers (positive and negative): > > ... -2, -1, 0, 1, 2, 3, ... reversed is > ... 3, 2, 1, 0, -1, -2, ... > > This even works for some uncountable sets (e.g. all the reals, or the > reals in a given interval). > > If your objection is more that infinite sets lack at least one "end", > remember that "infinite" means "without end", so the lack of an "end" is > inescapable. > > [snip]
"A totally ordered set (A,<=) is said to be well ordered (or have a well-founded order) iff every nonempty subset of A has a least element (Ciesielski 1997, p. 38; Moore 1982, p. 2; Rubin 1967, p. 159; Suppes 1972, p. 75). Every finite totally ordered set is well ordered. The set of integers Z, which has no least element, is an example of a set that is not well ordered."