> 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.
My caution is regarding the application to infinite sets of concepts taken from the realm of finite sets and assuming conclusions drawn from those applications are meaningful. The examples I gave were intended to demonstrate that some ideas which are clearly applicable to finite sets fail to be meaningful when applied to infinite sets. If we consider the set of integers rather than natural numbers, I ask what it means to rotate the members of that set to the left or right. Is zero the middle element of that set? If so, does that mean there are more numbers smaller than a billion than there are larger than a billion? There are plenty of other examples one might conjure up to show that such an infinite set is unlike a finite set.
I'm not rejecting the idea of infinite sets altogether, but I am also not willing to accept some of the assertions made about them without careful scrutiny.
Are you familiar with Cantor's theory of transfinite numbers? -- Nil conscire sibi