[FOM] Cardinals and Choice

[Robert Solovay]

The proposition that cardinals are comparable implies the axiom of
choice. Surely this must be in the treatise of Rubin and Rubin on the
axiom of choice.

Here is a sketch of the proof. Let X be a set.  Consider the
collection of all well-orderings of a subset of X, By comparability of
well-orderings they can be pieced together to get a longest
well-ordering such that every initial segment of it maps into X, but
it does not. (If need be, add a point on the end to achieve this.) So
we have a well-ordering that does not map injectively into X. By
comparability, X maps 1-1 into this well-ordering. Hence X can be
well-ordered.

--Bob Solovay

On Wed, Dec 22, 2010 at 5:51 AM, Richard Heck <rgheck at brown.edu> wrote:
>
> All the helpful books are at the office, so I'll ask here: Does the
> principle that the cardinals are well-ordered imply the axiom of choice?
> It clearly implies countable choice, right? (Since an infinite but
> Dedekind finite concept would allow us to create a descending sequence
> of cardinals.) Does it imply more?
>
> Richard
>
> --
> -----------------------
> Richard G Heck Jr
> Romeo Elton Professor of Natural Theology
> Brown University
>
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>