[FOM] Cardinals and Choice

T.Forster@dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Thu Dec 23 01:33:49 EST 2010

AC is equivalent to the assertion that cardinals are even merely *linearly* 
ordered (by magnitude).  As far as i know it is an open question whether or 
not AC is equivalent to the assertion that the natural order relation on 
cardinals is wellfounded.

On Dec 23 2010, Richard Heck 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?

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