Re: Two neat facts from set theory

From: Butch Malahide (bof_at_sunflower.com)
Date: 01/01/05 

Date: 1 Jan 2005 01:14:22 -0800

Stephen J. Herschkorn wrote:
> Butch Malahide wrote:
>
> > <>Stephen J. Herschkorn wrote:
> >
> >> <>2) Given two well-orderings of an infinite set X, there exists a
> >> subset Y of X such that Y has the same cardinality as X and the
> >> well-orderings agree on Y. I do not know if you can prove this
> >> without the axiom of choice.
> >
> >
> >It's quite easy to prove without the axiom of choice. To start with,
> >you can assume that the order type of each of your well-orderings is
> >the initial ordinal of cardinality |X|; i.e., each element has fewer
> >than |X| predecessors. Now, let x_0 be the least element with
respect
> >to the first well-ordering; delete x_0 and all elements that precede
> >x_0 in either ordering, and let x_1 be the least remaining element
with
> >respect to the first well-ordering; and so on.
>
> I don't follow this. Aren't you making a big step is in your initial

> assumption?

Just a couple of small steps. Let t be the least ordinal of cardinality
|X|.
Let X' be a subset of X having order type t with respect to the first
well-ordering, and let X'' be a subset of X' having order type t with
respect to the second well-ordering. So X'' has order type t with
respect to both well-orderings. Work with X'' instead of X.

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