
Re: Cantor's first proof in DETAILS
Posted:
Nov 29, 2012 11:16 PM


On Nov 29, 9:13 am, Marshall <marshall.spi...@gmail.com> wrote: > On Thursday, November 29, 2012 7:18:59 AM UTC8, 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 coimage 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: wellordered.
Regards,
Ross Finlayson

