[FOM] Cardinals and Choice
Robert Solovay
solovay at gmail.com
Thu Dec 23 05:37:06 EST 2010
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
>
>
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