Re: Well Ordering the Reals

Robert Low said:
> Tony Orlow wrote:
> > Robert Low said:
> >>If you keep the axiom of infinity, you have limit ordinals,
> >>since the axiom of infinity is saying precisely that one
> >>particular limit ordinal (omega) exists.
> > What it says, according to MathWorld, is that there exists a set that includes
> > the null set, and for every element in the set, the successor is in the set,
> > defining the Von Neuman ordinals.
>
> No, this does *not* define the von Neumann ordinals. It says that
> one particular class of von Neumann ordinals is a set.
>
> > What causes an inductive definition of a set
> > to imply the existence of elements outside the set? Basically, the idea is that
> > the size is always greater than the largest value,, and so the size of the set
> > of all finites is larger than all finites. But, that is easily remedied by
> > starting with {1} instead of {0}, and having the size EQUAL to the largest
> > value. It's a simple matter of definition and an error of 1 that requires limit
> > ordinals, as far as I can see. Limit ordinals are a cute trick but not a good
> > soluton.
>
> Sorry, but I'd struggle even to make up a story that makes sense
> using those words. Trying to extract sense from them in that order
> is completely beyond me. You seem to think that if we left 0 out
> of the set of ordinals then there would be no limit ordinals, but
> I can't imagine why you would think that. There would still be
> the set {1,2,3,....} and it would still be a limit ordinal, not
> a successor, and there still wouldn't be an element of the set
> which was the number of elements of it.
>
Ack! If you start with 0, then the number of elements in the set is always
GREATER than the largest value, as you add successors. If you start with 1
as your first natural, the number of elements is always EQUAL to the largest
element, and therefore never becomes infinite as long as the elements
themselves are finite.
I bet you could make up a neat story with those words, if you really tried.
--
Smiles,

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

Relevant Pages

  • Re: Well Ordering the Reals
    ... >> If we add the assumption that there are no limit ordinals then you've ... >> made set theory inconsistent. ... But ZF includes the axiom of infinity. ...
    (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, ... > Tony Orlow wrote: ... Limit ordinals are a cute trick but not a good ...
    (sci.math)
  • 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)
  • Re: Well Ordering the Reals
    ... >> Tony Orlow wrote: ... >> If we add the assumption that there are no limit ordinals then you've ... > You could do ZF with the axiom of infinity replaced by its negation. ... identifiable element has an identifiable predecessor. ...
    (sci.math)
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