Re: Well Ordering the Reals

William Hughes wrote:
> Tony Orlow wrote:
> > William Hughes said:
> > >
> > > Tony Orlow wrote:
> > > > William Hughes said:
> > > > >
> > > > > Which of the following is false?
> > > > >
> > > > > For evey n in N (the finite naturals) the
> > > > > set A= {1,2,3...,n} has size equal to the largest element in A
> > > > > Therefore the size of. N is the largest element in N.
> > > > >
> > > > > For evey n in N (the finite naturals) the
> > > > > set A= {1,2,3...,n} is bounded.
> > > > > Therefore N is bounded..
> > > > >
> > > > The second is false. The first is correct, despite the discomfort it may give
> > > > you.
> > >
> > > To be clear. You are stating the the first
> > > argument is correct (not only is the result correct but the
> > > agument is correct) and the second argument is wrong.
> > >
> > > Can you explain what the difference is?

> > Yes. In the second, we note the equivalence between the largest element and set
> > size for any initial set of naturals. Wherever we stop, value-wise, that is the
> > size of our set. If we don;t stop, we have no set size. When they are equal,
> > where one exists the other does, and where one doesn't the other doesn't
> > either.
> >
> > In the second, given the choice of an upper bound, we have an upper bound, but
> > without an upper bound we have no upper bound. For lack of an upper bound, we
> > have no largest element, and we have no set size.
>
> You seem to have missed the point. Both arguments are
> of the form
>
> For evey n in N (the finite naturals) the
> set A= {1,2,3...,n} has property X.
> Therefore N has property X.
>
>
> You are saying that if you replace X with "has a set size equal
> to the greatest element", you get a true argument, but
> if you replace X with "is bounded" you get a false argument. How
> can I tell which properties produce true arguments and
> which properties produce false arguments.

Oh come on, come on!! You're not imagining enough. Obviously the true
properties produces true arguments, and the false ones false arguments.
I mean, as Tony says, like, the set size is obviously equal to the
greatest element, and there's no way getting to infinity will prevent
that, whereas bounded means not reaching infinity, so obviously once
you get to the edge, I mean infinity, then nothing's bounded any more,
is it? I mean, you know, the sky's the limit for interplanetary travel,
and this is the same sort of thing.

Brian Chandler
Imagine, Imaginator, Imaginatest!
http://imaginatorium.org

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >> William Hughes said: ... Wherever we stop, value-wise, that is the ... In the second, given the choice of an upper bound, we have an upper bound, but ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > William Hughes said: ... >> Tony Orlow wrote: ... we note the equivalence between the largest element and set ... > In the second, given the choice of an upper bound, we have an upper bound, but ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >> Tony Orlow wrote: ... despite the discomfort it may give ... > In the second, given the choice of an upper bound, we have an upper bound, but ... For evey n in N (the finite naturals) the ...
    (sci.math)
  • Re: Orlow cardinality question
    ... > Tony Orlow (aeo6) wrote: ... >> upper bound, as you folks keep emphasizing when you ask for a largest finite ... and yet you claim it is infinite. ...
    (sci.math)