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 that, and it's false on the face of it. That's your bald lie, Hancher, and typical of them. Do you treat your other colleagues that way or do they have nothing to do with you? I find it offensive for you to tell or repeat lies about me, or others.
So, "no", think you.
And Dirac's delta is modeled by real functions, and quite suitable for use in real analysis, and quite used. 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.
And EF goes to one. 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."
There are only and everywhere real numbers, of the continuum, of real numbers, between zero and one.