```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.>> MarshallBasically this reads that d is unbounded and n ranges from zerothrough d.EF(n,d) is a family of functions, with d unbounded it's a particularEF(n) with properties modeled by those standard real functions, in asimilar way as to how, for example, Dirac's delta or Heaviside's stepare so modeled.Being dense in the reals, in its range, leads to a variety ofconsiderations of anywhere dense elements in their natural order.The reals:  well-ordered.Regards,Ross Finlayson
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