Re: Nice little total order problem.

From: William Elliot (marsh_at_privacy.net)
Date: 03/17/05 

Date: Thu, 17 Mar 2005 00:41:47 -0800

On Thu, 17 Mar 2005, Butch Malahide wrote:
>
> Stephen J. Herschkorn wrote:
>> Butch Malahide wrote:
>>> One of them is a set of order type eta times omega-one. That is, the
>>> set of all ordered pairs (a,r), where a is a countable ordinal and r
>>> is a rational number, ordered by first differences. In other words, a
>>> copy of the rational line is inserted between each countable ordinal
>>> and its successor.
>>>
>> I don't understand. Is this the same as the lexicographic ordering on
>> aleph1 x Q?
>
> I think so, but I was wrong in thinking that it would have a first
> element.
>
Yes, your 1st construction was exactly that.
No, your 2nd construction is, in order stated, a verbal description of
         0 + Q + 1 + Q + 2 + ...
or the lexicographical order of omega_1 x ([0,1) /\ Q)
which I think differs from lex omega_1 x Q only in having a first
element.

>> If so, how is the interval between (1,0) and (2,0) isomorphic to
> Q?
>
> That interval contains the pairs (1,x) where x is a positive rational
> number, followed by the pairs (2, y) where y is a negative rational
> number. That's a countable set, dense in itself, with no first or last
> element; so it's isomorphic to Q.
>
Exactly so, the cannonial description of the order type eta.