Re: Well Ordering the Reals

Robert Low said:
> MoeBlee wrote:
> > Tony Orlow wrote:
> >>If there are
> >>no limit ordinals
> > If we add the assumption that there are no limit ordinals then you've
> > made set theory inconsistent.
>
> You could do ZF with the axiom of infinity replaced by its negation.
> You wouldn't have limit ordinals then...
>
Bt that is just a statement of the inductive definition of an infinite set of
successions. That is the part I'd like to see preserved, such that every
identifiable element has an identifiable predecessor. Myabe I am
misunderstanding which part of the axiom you wish to negate? Certainly you
aren't suggesting a change to say such a set does not exist?
--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >>> If you keep the axiom of infinity, you have limit ordinals, ... > Tony Orlow wrote: ... Limit ordinals are a cute trick but not a good ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >> You could do ZF with the axiom of infinity replaced by its negation. ... >> You wouldn't have limit ordinals then... ... > Bt that is just a statement of the inductive definition of an infinite set of ... > identifiable element has an identifiable predecessor. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >>>If you keep the axiom of infinity, you have limit ordinals, ... Limit ordinals are a cute trick but not a good ...
    (sci.math)
  • Re: Question regarding limit ordinals and transfinite cardinals.
    ... The axiom of infinity is "not quite" an assumption? ... It does not prove the existence of any other limit ordinals. ... The existence of limit ordinals except w is hardly a direct corollary of the axiom of infinity, and indeed without AxC I suspect that you can't prove the existence of any limit ordinal except w. ...
    (sci.logic)
  • Re: Well Ordering the Reals
    ... >> If you keep the axiom of infinity, you have limit ordinals, ... > of all finites is larger than all finites. ... Limit ordinals are a cute trick but not a good ...
    (sci.math)
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