[FOM] Minor result. New, or known to Cantor?

BillOrEllen fetmarsh at olypen.com
Mon Jun 23 11:02:27 EDT 2003

```Working on ways to teach kids decimal order, I recently noticed that the
sets of words over any alphabet containing at least two letters are always
of the same order type, omega followed by eta copies of omega.  Once
noticed, it's easy to prove:  Let A be a totally ordered finite set of two
or more "letters" with first element a.  Let B be A-{a}.  Then A+ is the
disjoint union of {a}+ and
(A*)B({a}*).  Alphabetical order on A+ consists of an omega sequence for
{a}+ followed by an omega sequence for  every element of (A*)B.  (A*)B is of
type eta, i.e., a countable dense linear order without endpoints, as can be
proved directly or by looking at it as the set of base n "decimals" that
name exactly the rationals in (0,1) with denominator a power of n.

If this result is already known, I'd appreciale a reference for it.

Bill Marsh
204 Columbus Ave.
Port Angeles, WA 98362
fetmarsh at olypen.com
360-457-6758

```