On Sat, 28 Jun 2014, scattered wrote: > On Saturday, June 28, 2014 5:12:02 AM UTC-4, William Elliot wrote:
> > How is the trichotomy of cardinals equivalent to AxC? > > Defining the cardinals as done within ZF, as the > > the initial ordinals, aren't they well ordered? > > Thusly a proof for AxC? > > Cardinals are defined as equivalence classes under the equinumerable > equivalence relation. Without AC these equivalence classes can fail > trichotomy. In this case they are not automatically equivalent to > certain ordinals. > What set theory? NF, NBG? Certainly not ZF. How could such classes be defined as sets?