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>>>>>>>>>>> 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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