In article <see-6568E7.firstname.lastname@example.org>, Barb Knox <email@example.com> wrote:
> In article <NvudnSgQHKv-VyrZnZ2dnUVZ_tSdnZ2d@speakeasy.net>, > Hatto von Aquitanien <abbot@AugiaDives.hre> wrote: > > [snip] > > > I believe worrying about the existence of any given element of the set of > > all natural numbers is missing the point. The difference I perceive > > between infinite sets and finite sets is structural. Even if we to admit > > the concept of infinite sets into our weltanschauung, we should understand > > that they are structurally different from finite sets. 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]
On rereading, it seems more that your objection is to sets that are well-ordered by "<" but not also by ">" (which my examples fail to address, being totally ordered but not well-ordered). You like the fact that all totally-ordered finite sets are well-ordered in both directions; but the notion of well-ordering for infinite sets also has its uses.