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Re: Cantor's first proof in DETAILS
Posted:
Nov 29, 2012 11:16 PM
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On Nov 29, 9:13 am, Marshall <marshall.spi...@gmail.com> wrote:
> On Thursday, November 29, 2012 7:18:59 AM UTC-8, Ross A. Finlayson wrote:
> > On Nov 28, 4:58 pm, Marshall <marshall.spi...@gmail.com> wrote:
> > > Um, so EF is a restriction of division?
>
> > > The domain of x depends on the value of d. I don't recall having seen
> > > that sort of thing before, but I guess I do know what that means.
> > > But I can't figure out what the domain of d is. It sorta looks like the
> > > domain of d depends on what d is, but what the heck would that mean?
>
> > > And it's just a name, but what about EF has anything to do with
> > > equivalency?
>
> > > Marshall
>
> > Mr. Spight, it's about the equivalency or equipollency or equipotency
> > of infinite sets.
>
> Ok.
>
> > EF(n) = n/d, d->oo, n->d.
>
> I'm not very good with limits, so I'm not sure exactly what this means.
> Are you saying that the value of EF is a limit? A double limit? Also
> I'm not sure how n can approach d when n is a parameter of EF.
>
> I guess also you're saying EF is *not* a restriction on division.
>
> > Properties include:
> > EF(0) = 0
> > EF(d) = 1
> > EF(n) < EF(n+1)
> > The domain of the function is of those natural integers 0 <= n <= d.
>
> Whoa, you lost me. Here, EF has only one parameter, and you show
> 1. EF(0) = 0
> 2. EF(d) = 1
> but also
> 3. EF(n) < EF(n+1)
>
> But it seems you have done some hidden binding of d that I'm not
> clear about. Given 2. above, EF applied to any nonzero number yields
> 1, which contradicts your 3. above. What gives?
>
> > It's very simple this. Then, not a real function, it's standardly
> > modeled by real functions:
> > EF(n,d) = n/d, d E N, n->d
> > with each having those same properties.
>
> > Then, the co-image is R[0,1] as is the range.
>
> I'm not following.
>
> Marshall
Basically this reads that d is unbounded and n ranges from zero
through d.
EF(n,d) is a family of functions, with d unbounded it's a particular
EF(n) with properties modeled by those standard real functions, in a
similar way as to how, for example, Dirac's delta or Heaviside's step
are so modeled.
Being dense in the reals, in its range, leads to a variety of
considerations of anywhere dense elements in their natural order.
The reals: well-ordered.
Regards,
Ross Finlayson
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