[continued from previous message]   
      
   >>>>>> had something about Kosmanson's concerns about handling infinities.   
   >>>>>>   
   >>>>>> Here I quote the part of the blog that contained that problem:   
   >>>>>>   
   >>>>>> (beginning of the quote)   
   >>>>>>   
   >>>>>>   
   >>>>>>    "Then, swoooooooshhshsh!.... and Jesus and all that intense light   
   >>>>>> went   
   >>>>>> back up and out of there. Physfit looked up and there wasn't even an   
   >>>>>> opening in the ceiling anymore. But now for some reason he was   
   >>>>>> horizontally on the floor, in his bed. Right in the living room!   
   >>>>>>   
   >>>>>> He thought a bit about what was happening, when he found himself   
   >>>>>> quite   
   >>>>>> hungry. Last time he had eaten anything was the night before he had   
   >>>>>> waken up on the summit of the magic mountain in an urban Dallas area.   
   >>>>>>   
   >>>>>> He thought to himself, "I'm going to assume that more than 48   
   >>>>>> hours has   
   >>>>>> passed since. So got up and walked to the kitchen and took a look   
   >>>>>> inside   
   >>>>>> refrigerator. There was nothing there but the cat food he had   
   >>>>>> cooked on   
   >>>>>> the day he first saw the magic mountain. He got on the computer to   
   >>>>>> order   
   >>>>>> something zesty from HelloFresh. After choosing the closest to a   
   >>>>>> healthy   
   >>>>>> nice pre-agricultural food kit, he clicked, "Go to checkout" button,   
   >>>>>> after which the computer waited for a few seconds but instead of   
   >>>>>> getting   
   >>>>>> to the check out screen, a screen came up to make sure Physfit was   
   >>>>>> not a   
   >>>>>> robot. It had a simple question that he had to give it the correct   
   >>>>>> answer, otherwise food nommo.   
   >>>>>>   
   >>>>>> The question went like this:   
   >>>>>>   
   >>>>>>      "In math, is there a difference between the two numbers   
   >>>>>> 0.999999...   
   >>>>>> and 1 ?"   
   >>>>>>   
   >>>>>> The digits of "9" continued forever to the right of the radix   
   >>>>>> point. So   
   >>>>>> of course, Physfit clicked on the "yes" button. If there was not a   
   >>>>>> difference, then one wouldn't even bother to write 1 in that funky   
   >>>>>> form,   
   >>>>>> using an infinite series of digit 9.   
   >>>>>>   
   >>>>>> But the screen disappeared, and a message said, "You're a robot.   
   >>>>>> Bye!"   
   >>>>>>   
   >>>>>> Physfit said, "Fuck!" (first of the fix number of curses Jesus had   
   >>>>>> allowed him for that day). So he took a pen and paper and started   
   >>>>>> jotting down:   
   >>>>>>   
   >>>>>>      x = 0.99999....   
   >>>>>>   
   >>>>>> Therefore:   
   >>>>>>   
   >>>>>>      10x = 9.99999....   
   >>>>>>   
   >>>>>> Now he subtracted the former from the latter:   
   >>>>>>   
   >>>>>>      10x - x = 9.99999... - 0.99999...   
   >>>>>>   
   >>>>>> Which simplifies to:   
   >>>>>>   
   >>>>>>      9x = 9   
   >>>>>>   
   >>>>>> And therefore:   
   >>>>>>   
   >>>>>>      x = 1   
   >>>>>>   
   >>>>>> "What the fuck??", said Physfit (his 2nd curse of the day).   
   >>>>>>   
   >>>>>> Why x which was 0.99999... and not 1, turned out to be 1? ... "   
   >>>>>>   
   >>>>>>   
   >>>>>> (end of quote)   
   >>>>>>   
   >>>>>>   
   >>>>>> So, is this problem pointing to what Kosmanson has been so keen   
   >>>>>> about? :)   
   >>>>>>   
   >>>>>>   
   >>>>>>   
   >>>>>>   
   >>>>>>   
   >>>>>   
   >>>>> Once I was reading a book or article,   
   >>>>> and was introduced the introduction of .999 (...),   
   >>>>> vis-a-vis, 1. A cohort of subjects was surveyed   
   >>>>> their opinion and belief whether .999, dot dot dot,   
   >>>>> was equal to, or less than, one. About half said   
   >>>>> same and about half said different.   
   >>>>>   
   >>>>>   
   >>>>> It's two different natural notations that happen   
   >>>>> to collide and thus result being ambiguous.   
   >>>>>   
   >>>>> So, then these days we have the laws of arithmetic   
   >>>>> introduced in primary school, usually kindergarten,   
   >>>>> about the operations on numbers, and also inequalities,   
   >>>>> and the order in numbers.   
   >>>>>   
   >>>>> Yet, even the usual account of addition and its   
   >>>>> inverse and its recursion and that's inverse,   
   >>>>> as operators, of whole numbers, has a different   
   >>>>> account, of increment on the one side, and, division   
   >>>>> on the other, sort of like the Egyptians only had   
   >>>>> division or fractions and Egyptian fractions,   
   >>>>> and tally marks are only increment, that though   
   >>>>> it was the Egyptian fractions that gave them a   
   >>>>> mathematics, beyond the simplest sort of conflation   
   >>>>> of "numbering" and "counting".   
   >>>>>   
   >>>>> So, where ".999 vis-a-vis 1" has a deconstructive account,   
   >>>>> to eliminate its ambiguities with respect to what it's   
   >>>>> to model, or the clock-arithmetic and field-arithmetic,   
   >>>>> even arithmetic has a deconstructive account, then,   
   >>>>> even numbering versus counting has a deconstructive account,   
   >>>>> to help eliminate what are the usually ignored ambiguities.   
   >>>>>   
   >>>>>   
   >>>>> So, pre-calculus, the course, goes to eliminate or talk   
   >>>>> away the case .999, dot dot dot, different 1. Yet,   
   >>>>> it can be reconstrued and reconstructed, on its own   
   >>>>> constructive account. So, it's a convention.   
   >>>>>   
   >>>>>   
   >>>>> It's "multiplicity theory", see, that any, "singularity   
   >>>>> theory", which results as of admitting only the principal   
   >>>>> 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   
      
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