
Re: Cantor's first proof in DETAILS
Posted:
Dec 3, 2012 10:11 PM


On Dec 3, 2:44 pm, Virgil <vir...@ligriv.com> wrote: > In article > <f40c3a4a0954451bbb0e58e14bf7b...@q5g2000pbk.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > > > > > > On Dec 2, 10:39 pm, Virgil <vir...@ligriv.com> wrote: > > > In article > > > <768a7f472e2340c2a27a1483f5b65...@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 twopoint 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 iotavalues, 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 that > > Of 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 functions > > While 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 deltaepsilonics 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 > > 
Yes, Dirac's delta as it is: is extra the standard, encompassing it, AND beyond it. (And of course it's rigorously modeled by standard real functions and relevant to the pure AND applied in mathematics.) And Heaviside's step simply has each value between [1,1] in y on the vertical line through the origin.
And EF goes to one. So though none of the _finite_ naturals have that n/d > eps, for each positive eps < 1 there is an element of the range y thus that y > eps, or y /> 1, as it does: y = EF(n) > 1. So, as is typical, you didn't complete the statement. Would you have that lim Sum 1/2^n < 2, n e N? 0 e N: lim Sum_n=0^\infty 1/2^n = 2. It's not that there's no smallest n that it is, it's that there's no largest n that it can't be.
So, it is false for you to (m)utter that d/d =/= 1, or that your false challenge applies to any. It's not a failure to note that your "challenge" is to confirm a falsehood, nor is it much of a challenge.
It's not that there's no smallest n that it is, it's that there's no largest n that it can't be.
There are only, and everywhere, elements of the "real" continuum, as we well know them as real numbers, between zero and one, and so are those from infinity back to infinity fore, for some including infinity, and for none including transfinite cardinals nor your imaginary numbers (which are elements of the Argand plane and in extension to the hypercomplex).
No, it is our rules, of the reals. And, those are discoveries, not inventions, for the real, to the concrete, in mathematics, with rigor. We are simply curators, not owners, of these eternal truths.
And, application already has and uses the projectively extended real numbers, in mathematics, for, for example, physics.
Then it's not so much a challenge as a quest: where are our applications of transfinite cardinals? And, if there's a universe, mathematically, containing, itself, isn't that on the face of it a reasoning as to why: mathematics should consider at least such a comprehensive totality?
Because it is total.
And to my colleagues, and generally, warm regards,
Ross Finlayson

