```Date: Dec 3, 2012 5:44 PM
Author: Virgil
Subject: Re: Cantor's first proof in DETAILS

In article <f40c3a4a-0954-451b-bb0e-58e14bf7b6bc@q5g2000pbk.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:> On Dec 2, 10:39 pm, Virgil <vir...@ligriv.com> wrote:> > In article> > <768a7f47-2e23-40c2-a27a-1483f5b65...@qi10g2000pbb.googlegroups.com>,> >  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:> >> > > Card(N) isn't a real quantity.> >> > Neither, in a very primitive sense, are 1, 2 and 3. The naturals are> > precursors of both the integers and the positive rationals, both of> > which are precursor to the rational field  which is itself a precursor> > to the reals.> >> > But Card(N) is  real property, i.e., one which really exists.> >> > > So what you say there is wrong, though> > > it's wrong twice.> >> > Only when I copy what you say.> >> > > Though if you're interested in the real point at> > > infinity, well, you should be able to find description of number-> > > theory's point at infinity or one- or two-point compactification of> > > the reals and integers (in the projectively extended real numbers).> > > Cardinals are defined by themselves, don't be putting them where they> > > don't go, those aren't compatible types.> >> > Are you claiming that one cannot speak of the cardinality of the> > naturals, or rationals, or reals? That you may not know how does not> > mean others are all equally ignorant.> >> >> >> > > There are only and everywhere real numbers between zero and one.> >> > There cannot be 'only and everywhere' numbers that are constrained not> > to be anywhere except between zero and one.> >> > > Here, 0 < EF(1) < 1.> >> > But according to Ross very own definition of his alleged EF, one can> > show that for every n in |N and every positive epsilon, that EF(n) <> > epsilon.> >> > >  The arithmetic of iota-values, representing values> > > from the continuum, of real numbers, is different for the operations> > > as addition, and multiplication, simply as repeated addition.> >> > Repeated addition, adding two numbers then adding the third , then the> > fourth, etc.,  works fine, though a bit tediously, for all number> > systems contained in the reals. Or even contained in the complexes or> > quaternions or octonions, or vector spaces, for that mater..> >> >> >> > > Dirac's delta is regularly used in real analysis, for example in the> > > solutions of differential equations.> >> > When it is used, the analysis is not quite real as it is not a real> > function.> >> > >  Heaviside's step can be seen as continuous> >> > By what definition of continuous? One may chose to ignore its> > discontinuity, but that does not make it continuous.> >> > , it just is horizontal from the left, vertical at the> >> > > origin> >> > But how is  function which takes the value 0 at 0 and the value 1 at> > every positive real argument and -1 at every negative real argument> > satisfy the INTERMEDIATE VALUE theorem on any interval containing 0 as> > an interior point? or even the mean value theorem?> >> > (the INTERMEDIATE VALUE theorem says that a function continuous on any> > interval [a,b] must assume every value between f(a) and f(b) at some> > point of that interval)> >> > > No, these are considerations of the plain mathematical universe shared> > > among us, using standard definitions and working toward conciliation> > > of intuition and rigor, thank you.> >> > But Ross' mathematical universe does not use, much less conform to,> > standard definitions, or even allow them, and he opposes anything> > resembling rigor.> > --> > > Hancher writes to our public forum on mathematics:  "But Ross'> mathematical universe does not use, much less conform to, standard> definitions, or even allow them, and he opposes anything resembling> rigor. "> > I dispute thatOf course your do! WM also disputes his critic's views. > and it's false on the face of it.If it were, you should be able to prove it, using standard mathematics.That you haven't guts your claim.> That's your bald> lie, Hancher, and typical of them.To call it a lie is easy, to prove it is a lie is apparently enough beyond Ross' powers that he does not even try.> Do you treat your other colleagues> that way or do they have nothing to do with you?I treat colleagues with the respect they deserve, but merely posting to sci.math does not establish collegiality. >  I find it offensive> for you to tell or repeat lies about me, or others.I find it AT LEAST equally offensive when you lie about me, or others.> > So, "no", think you.> > No, "thunk".> > (Shrug.)> > And Dirac's delta is modeled by real functionsWhile the Dirac Delta is certainly expressible as a limit of a sequence of continuous real functions defined on all of R, and specifically  continuous function at 0, it is not a uniform limit and thus need not be, and is not, even a function at all on all of R. A real function defined at 0 has a real number as its value at 0, but the Dirac Delta does not have a real number value at 0.>  And Heaviside's step is continuous where it is so defined >, and it's the same function with> regards to analysis, placing for the point discontinuity a connection,> obviously enough continuous and here satisfying the IVT. The Heaviside function has values  -1 at -1 and +1 at +1, so if it satisfies the intermediate value theorem, there must be some x between -1 and +1 at which the function has the value 1/pi.But it does not!> > And EF goes to one.Your EF should go to the garbage heap.> Here you mention delta-epsilonics and I'm quite> happy to work it up in that, re: density in R_[0,1], then,> continuity:  "Standardly modeled by standard real functions."You claim that there is some positive real value y such that EF(n) = y for all n in |N, and such that Sum_{n in |N} EF(n) = 1. Clearly any such y must be greater than zero, so there must be a smallest n in |N such that 1/2^n < EF(n), and then also infinitely many larger n's for which 1/2^n < EF(n) is true.So I challenge Ross to find any explicit natural number n, of which there must be infinitely many,  for which 1/2^n < EF(n) is true.1/10^n < EF(m) for any m in |N.Ross failure to do so will be justifiably taken as  evidence of his, and everyone's, inability to do so, and thus the falsity of hi clims.> > There are only and everywhere real numbers, of the continuum, of real> numbers, between zero and one.I can find lots of real numbers other than those between 0 an 1.> > And the real numbers:  they're not yours to keep.It is their rules that I keeps, and Ross does not.> > No, thank you,> > Ross Finlayson--
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