At 10:59 +0200 1999/04/29, Bjorn Lisper wrote:
>Just a final comment on total orders on sets: this makes sense, as regards
>operations where the order is important for the semantics, only if the
>elements of the set are drawn from an enumerable set. It would not be very
>sensible to, for instance, try to impose a total order on a set of
>functions. A total order of the objects representing the functions would of
>course be possible and could be used to have the balanced trees, but it
>would not be possible to, say, have a well-defined fold over that kind of
>set (or any indexed structure using this set as index set).
I am not sure I understand this reasoning with the fold here.
But in a math, by the axiom of choice, any set can be givem not only a
total order, but can be made to be well ordered, that is, every non-empty
set has a least element. Further, such a set can be exhausted by a repeat
of two operations, by taking the successor, or by a transfinite operation,
passing to higher ordinals.
Using a well ordered set, that seems to be possible.
Hans Aberg
* Email: Hans Aberg <mailto:[EMAIL PROTECTED]>
* Home Page: <http://www.matematik.su.se/~haberg/>
* AMS member listing: <http://www.ams.org/cml/>
