[FOM] Axioms that imply AC

William Tait wwtx at earthlink.net
Fri Feb 3 13:16:08 EST 2006

The statement that there is a well-ordering of the universe (i.e. a  
universal choice function) is an example. For example, this is true  
in L. It implies that set theory can be coded as a theory of  
ordinals. Ronald Jensen [BSL 1 No. 4 (1995)] referred to this  
possibility as the 'Pythagorean' conception of set theory:  
'Everything is number'.

Bill Tait

On Feb 3, 2006, at 1:33 AM, Andrej Bauer wrote:

> I student of mine is writing an undergraduate thesis on theorems in
> mathematics which are equivalent to the axiom of choice, which got me
> thinking: are there (relatively well-known) statements in  
> mathematics, or
> specifically set theory, which are _stronger_ than the axiom of  
> choice?
> Some form of choice for classes perhaps? What use does it have?  
> There are of
> course many weak forms of choice.
> Andrej Bauer
> University of Ljubljana
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