Re: Distinct linear orderings on Z

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/18/05 

Date: Fri, 18 Mar 2005 14:46:19 -0500

Wolf Kirchmeir said:
> Tony Orlow (aeo6) wrote:
> > Herman Rubin said:
> >
> >>In article <ot9_d.19071$aj7.16108@fe12.lga>,
> >>Stephen J. Herschkorn <sjherschko@netscape.net> wrote:
> >>
> >>>robert j. kolker wrote:
> >>
> >>>>Now here is a serious question: (therefore not addressed to Tony or
> >>>>Albert). How many (in the sense of cardinality) distinct linear
> >>>>orderings are there on Z. I say two linear orderings L1 and L2 are
> >>>>distinct if L1 cannot be gotten from L2 by an order isomorphism with
> >>>>the ground set Z.
> >>
> >>
> >>>I think the answer is aleph1 (the first uncountable ordinal). There are
> >>>certainly aleph1 nonisomprophic well-orderings (for aleph1 is the set of
> >>>all countable ordinals). Isn't any linear ordering of Z a pasting of
> >>>a reverse well-ordering to a well-ordering? Or can one show otherwise
> >>>that there are at most aleph1 linear orderings?
> >>
> >>The number of linear orderings is 2^(aleph_0). This is
> >>certainly an upper bound as the number of ordered pairs
> >>is countable. However, I do not know how many isomorphism
> >>classes there are, but it is at least 2^(aleph_0).
> >>
> >>To show it is at least 2^(aleph_0), divide Z into a
> >>countable family of countable sets. Keep these in
> >>order. For each of these sets, either supply an initial
> >>well-ordered component of type omega or of type omega^2,
> >>followed by a reverse ordering of type omega. So we have
> >>2^(aleph_0) distinct types of linear orderings.
> >>
> >>
> >
> > Is 2^(aleph_0) really idfferent from aleph_1? Certainly Aleph_0^2 is not aleph_
> > 1, but what is the relationship, really, between aleph_0 and aleph_1? How do
> > you define aleph_1 in terms of aleph_0? Wouldn't it be basically an exponential
> > relationship such as 2^aleph_0? Just a novice asking......thoughts?
>
> Aleph_1 is the cardinality of the set of all curves (lines).
>
The set of all lines in what space? 2D? Does it matter? If the lines are in 3D
is the cardinality aleph_2, or the integral of aleph_1? Excuse my ignorance. 

-- 
Smiles,
Tony

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