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


On Nov 29, 7:35 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > "Ross A. Finlayson" <ross.finlay...@gmail.com> writes: > > > > > > > > > > > On Nov 28, 4:58 pm, Marshall <marshall.spi...@gmail.com> wrote: > >> On Monday, November 26, 2012 11:33:02 PM UTC8, Virgil wrote: > > >> > I find a citation from r 9/22/99 In which Ross states, what may well be > >> > Ross' original "definition" of his alleged "Equivalency Function" which > >> > as any mathematician can plainly see is not a function at all, and is > >> > only equivalent to nonsense:: > > >> > " Consider the function > >> > f(x, d)= x/d > >> > for x and d in N. The domain of x is N from zero to d and the domain of > >> > d is N as d goes to > >> > infinity, d being greater than or equal to one. > >> > I term this the Equivalency Function, and note it EF(x,d), also EF(x), > >> > assuming d goes to > >> > infinity." > > >> >http://groups.google.com/group/sci.math/msg/af29323d694cf89e1999 > >> > "Equivalency Function" > > >> 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. > > EF(n) = n/d, d>oo, n>d. > > > 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. > > > 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. > > Is this a version of the natural density of a subset of the natural > numbers? > > http://en.wikipedia.org/wiki/Natural_density > > > Regards, > > > Ross Finlayson > >  > Alan Smaill
No, though half of the integers are even.
They're related concepts in establishing that, to the asymptotic, it can be established the size of proper subsets of the integers, relative to the integers. Half of the integers are even. It was wrong to say that it is not so, that not "half of the integers are even", and that in terms of other densities of proper subsets of the integers, that they aren't comparable. Those numbers have those properties. Then I was of the impression that Schnirelmann density from our number theory and asymptotic density were the same.
Here having infinitely many elements between zero and one, given their constant difference, is enough to have them be dense as in the reals.
Then density as a quantity and density as a property aren't the same thing, yet, they both are to denote the propensity of elements within others of theirs.
Half of the integers are even.
http://groups.google.com/groups/search?as_q=&as_epq=Half+of+the+integers+are+even
And when I noted that another had concurred with this with regards, to sets, it was roundly derided until they were directed to his Ph.D. at M.I.T.
Regards,
Ross Finlayson



