[continued from previous message]   
      
   >>>>>>>>>>> branch of otherwise a "bifurcation" or "opening" or   
   >>>>>>>>>>> "catastrophe"   
   >>>>>>>>>>> or "perestroika (opening)", as they are called in mathematics,   
   >>>>>>>>>>> branches, that singularity theory is a multiplicity theory,   
   >>>>>>>>>>> yet the usual account has that it's just nothing,   
   >>>>>>>>>>> or that it's apeiron and asymptotic.   
   >>>>>>>>>>>   
   >>>>>>>>>>>   
   >>>>>>>>>>> So, there's a clock arithmetic where there's a reason why   
   >>>>>>>>>>> that there's a .999, dot dot dot, _before_ 1.0, in the   
   >>>>>>>>>>> course of passage of values from 0, to 1, and, it's also   
   >>>>>>>>>>> rather particularly only between 0 and 1, as what results   
   >>>>>>>>>>> thusly a whole, with regards to relating it to the modularity   
   >>>>>>>>>>> of integers, the integral moduli.   
   >>>>>>>>>>>   
   >>>>>>>>>>> Thusly, real infinity has itself correctly and constructively   
   >>>>>>>>>>> back in numbers for "standard infinitesimals" here called   
   >>>>>>>>>>> "iota-values".   
   >>>>>>>>>>>   
   >>>>>>>>>>> Then, this is totally simple and looks like f(n) = n/d,   
   >>>>>>>>>>> for n goes from zero to d and d goes to infinity, this   
   >>>>>>>>>>> is a limit of functions for this function which is not-   
   >>>>>>>>>>> a- real- function yet is a nonstandard function and that   
   >>>>>>>>>>> has real analytical character, it's a discrete function   
   >>>>>>>>>>> that's integrable and whose integral equals 1, it illustrates   
   >>>>>>>>>>> a doubling-space according to measure theory in the measure   
   >>>>>>>>>>> problem,   
   >>>>>>>>>>> it's its own anti-derivative so all the tricks about the   
   >>>>>>>>>>> exponential   
   >>>>>>>>>>> function in functional analysis have their usual methods   
   >>>>>>>>>>> about it,   
   >>>>>>>>>>> it's also a pdf and CDF of the natural integers at uniform   
   >>>>>>>>>>> random,   
   >>>>>>>>>>> of which there are others, because there are at least three laws   
   >>>>>>>>>>> of large numbers, at least three Cantor spaces, at least three   
   >>>>>>>>>>> models of continuous domains, and, at least three probability   
   >>>>>>>>>>> distributions of the naturals at uniform random.   
   >>>>>>>>>>>   
   >>>>>>>>>>> So, "iota-values" are not the same thing as the raw   
   >>>>>>>>>>> differential,   
   >>>>>>>>>>> which differential analysts will be very familiar with as   
   >>>>>>>>>>> usually   
   >>>>>>>>>>> not- the- raw- differential yet only as under the integral bar   
   >>>>>>>>>>> in the formalism, yet representing about the solidus or   
   >>>>>>>>>>> divisor bar   
   >>>>>>>>>>> the relation of two quantities algebraically, then indeed   
   >>>>>>>>>>> there's   
   >>>>>>>>>>> that "iota-values" are as of some "standard infinitesimals", yet   
   >>>>>>>>>>> only under the limit of function the "natural/unit equivalency   
   >>>>>>>>>>> function"   
   >>>>>>>>>>> the N/U EF, about [0,1]. This thus results a model of   
   >>>>>>>>>>> a continuous domain "line reals" to go along with the usual   
   >>>>>>>>>>> standard   
   >>>>>>>>>>> linear curriculum's "field reals" then furthermore later there's   
   >>>>>>>>>>> a "signal reals" of at least these three models of continuous   
   >>>>>>>>>>> domains.   
   >>>>>>>>>>>   
   >>>>>>>>>>>   
   >>>>>>>>>>> The usual demonstration after introducing the repeating terminus   
   >>>>>>>>>>> and using algebra to demonstrate a fact about arithmetic,   
   >>>>>>>>>>> is good for itself, and is one of the primary simplifications   
   >>>>>>>>>>> of the linear curriculum, yet as a notation, it's natural that   
   >>>>>>>>>>> two different systems of notation can see it variously, then   
   >>>>>>>>>>> that it merely demands a sort of book-keeping, to   
   >>>>>>>>>>> disambiguate it.   
   >>>>>>>>>>>   
   >>>>>>>>>>> If you ever wonder why mathematics didn't have one of these,   
   >>>>>>>>>>> or, two of these as it were together, it does, and it's only   
   >>>>>>>>>>> a particular field of mathematics sort of absent the   
   >>>>>>>>>>> super-classical   
   >>>>>>>>>>> and infinitary reasoning, that doesn't.   
   >>>>>>>>>>>   
   >>>>>>>>>>> Then at least we got particle/wave duality as super-classical,   
   >>>>>>>>>>> then Zeno's classical expositions of the super-classical were   
   >>>>>>>>>>> just given as that the infinite limit as introduced in   
   >>>>>>>>>>> pre-calculus   
   >>>>>>>>>>> said we could ignore the deductive result that it really must   
   >>>>>>>>>>> complete,   
   >>>>>>>>>>> the geometric series.   
   >>>>>>>>>>   
   >>>>>>>>>> Then again, one can define the reals as the convergences of   
   >>>>>>>>>> uncountably infinitely many infinite series. There is no   
   >>>>>>>>>> differece   
   >>>>>>>>>> between 0.999... and 1, they are simply two different   
   >>>>>>>>>> representations   
   >>>>>>>>>> of the same mathematical object.   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>> Bullshit. The point in question is exactly whether what you say is   
   >>>>>>>>> bullshit :)   
   >>>>>>>>>   
   >>>>>>>>> The answer to the baby problem shows, quite simply, that X is   
   >>>>>>>>> indeed   
   >>>>>>>>> 0.9999... and _certainly_ not 1.   
   >>>>>>>>>   
   >>>>>>>>> Physics, only in its most useful form for humans, can speak for   
   >>>>>>>>> mathematics (that's where 1 + 1 equals 2 comes from - from direct   
   >>>>>>>>> observation by humans); and mathematics in general does not   
   >>>>>>>>> speak for   
   >>>>>>>>> physics at any level, for human or for future superhumans and AI   
   >>>>>>>>> all. It   
   >>>>>>>>> is only rarely used when techniques developed in math would help   
   >>>>>>>>> physics   
   >>>>>>>>> in its use for humans to eventually solve problems, again for   
   >>>>>>>>> humans.   
   >>>>>>>>>   
   >>>>>>>>> If you need help seeing the above baby problem's answer, then   
   >>>>>>>>> beg for   
   >>>>>>>>> it :)   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>   
   >>>>>>>> No, don't be making problems when there's a mis-understanding.   
   >>>>>>>>   
   >>>>>>>> It is so that the modern model of real numbers is as of   
   >>>>>>>> "equivalence classes of sequences with the property of being   
   >>>>>>>> Cauchy",   
   >>>>>>>> then as with regards to whether both least-upper-bound property and   
   >>>>>>>> measure 1.0 are stipulated rather than derived, has that here it's   
   >>>>>>>> acknolwedged that LUB is stipulated and measure 1.0 is stipulated   
   >>>>>>>> with regards to the objects of analysis meeting the objects of   
   >>>>>>>> geometry,   
   >>>>>>>> where for example Hilbert says "there must be a postulate   
   >>>>>>>> of continuity" as with regards to Leibniz' "there _is_ a principle   
   >>>>>>>> of perfection".   
   >>>>>>>>   
   >>>>>>>> Then, Dedekind is considered a sort of mere hanger-on and it's so   
   >>>>>>>> that models of reals as Dedekind cuts are considered shallow and   
   >>>>>>>> as after an assignment that presumes what it intends to   
   >>>>>>>> demonstrate.   
   >>>>>>>>   
   >>>>>>>> Two wrongs is two wrongs.   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>>   
   >>>>>>>   
   >>>>>>> But I (and my past audience in that linux newsgroup) am not that   
   >>>>>>> concerned to go that much down into the nitty gritty of this   
   >>>>>>> thing. The   
   >>>>>>> point in my blog there was to test the audience whether they were   
   >>>>>>> actually "programmers" like a programmer really is, or they were   
   >>>>>>> mere   
   >>>>>>> "code monkeys" hired by real programmers, to receive the menial   
   >>>>>>> parts of   
   >>>>>>> work, yet coming in the scene here in usenet pretending to be   
   >>>>>>> programmers. This was the whole point of that blog.   
   >>>>>>>   
   >>>>>>> And only one among them, Farley Flud, proved to be a real   
   >>>>>>> programmer. I   
   >>>>>>> understood that by watching how he _tackles_ these baby problems.   
   >>>>>>> Nobody   
   >>>>>>> else there, including many "engineers" and "computer scientists"   
   >>>>>>> there   
   >>>>>>> were actually programmers.   
   >>>>>>>   
   >>>>>>> That's the level at which my baby problem was posed. I have not   
   >>>>>>> delved   
      
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