On 12/9/2012 12:30 PM, WM wrote: > On 9 Dez., 17:24, fom <fomJ...@nyms.net> wrote: >> On 12/9/2012 3:20 AM, WM wrote:
>> So, why is there no global axiom of choice? > > As far as I am informed, *the* axiom of choice is global. There is no > exception. Zermelo proved: Every set can be well-ordered. >>
The axiom of choice only applies to sets within a given model.
Consequently, for any set model whose domain is obtained from a larger model, the axiom of choice holds from the larger model.
But, the constructible universe is a class model.
Models obtained from the constructible universe by forcing methods need not have a one-to-one correspondence with the ordinals as defined in their scope. The axiom of choice only applies to the objects of the domain.
My question speaks to what may be considered as admissible as a model in such cases.