Virgil
Posts:
9,012
Registered:
1/6/11


Re: Cantor's first proof in DETAILS
Posted:
Dec 3, 2012 5:44 PM


In article <f40c3a4a0954451bbb0e58e14bf7b6bc@q5g2000pbk.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson@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 

