Date: Nov 29, 2012 11:16 PM
Author: ross.finlayson@gmail.com
Subject: Re: Cantor's first proof in DETAILS
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