[FOM] Correction to problem on order types
Jeremy Clark
jeremy.clark at wanadoo.fr
Sun Mar 6 16:21:07 EST 2005
I don't know if this is the error Joe had in mind, but it did occur to
me after I submitted an answer that there are in fact many more order
types satisfying this condition. Take a set S \subset \omega_1. Then
define A_x to be Q (order type of rationals) if x \in S and 1+Q
otherwise. Let A_S = sum A_x for x \in \omega_1. I *think* one can show
something like: if S \setminus T is stationary in \omega_1 (meets every
closed unbounded set in \omega_1) then A_S cannot be order isomorphic
to A_T. It is possible to define 2^\omega_1 sets such that any two of
them have the property that their setwise difference is stationary. So
it follows that there are 2^\omega_1 sets satisfying Joe's condition.
regards,
Jeremy Clark
On Mar 6, 2005, at 7:05 am, Robert M. Solovay wrote:
> Joe,
>
> Can you elaborate what the error is? I got eleven as well. The
> empty set; the one point set and then 3 times 3 where the possibilities
> are open, closed, or "the long line" for each end.
>
> --Bob Solovay
>
>
>
> On Sat, 5 Mar 2005 JoeShipman at aol.com wrote:
>
>> In the statement of the problem, I should require that every open
>> interval (a,b) with a<b is isomophic to the real numbers, not
>> rational numbers.
>>
>> That was the form I had originally come up with the problem, but then
>> I got too clever. Dave Marker claims that changing "reals" to
>> "rationals" allows more solutions than I had contemplated, and
>> pointed out why my comtemplated proof for "rationals" fails.
>>
>> -- JS
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