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Subject: Re: How do you have a number line without a first quantity closest to
 zero?
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 28 Jul 2024 09:02:26 -0700
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On 04/05/2020 06:48 PM, Ross A. Finlayson wrote:
> On Sunday, April 5, 2020 at 4:54:13 PM UTC-7, Michael Moroney wrote:
>> Mitch Raemsch <mitchrae3323@gmail.com> writes:
>>
>>> Zero no quantity then first quantity on the number line...
>>> one divided by the unlimited.
>>
>> And now I draw a point halfway between zero and your point, which is no longer
>> any sort of "first quantity".  So there.
>
> The metrizing ultrafilter has a countable aspect
> that reflects all the analytical character of the
> real function under countable additivity.
>
> (For measure theory.)
>
> The usual notion of differential patches, regions, or areas,
> as sequential and each having a next, is actually a property
> of continuity established for example by finding a smaller one.
>
> I.e. it's for a usual definition of continuous function.
>
> It's unfair to differential calculus
> and Leibniz' summation notation
> for the integral bar
> to not have the "differential"
> (for: differences).
>
> I.e., definite integration is always about the bounds
> and for also where there are no bounds.
>
> I.e., all functions are also piece-wise.
>
> Having a function that ranges from zero to one
> in constant differences instead of geometric series
> or the usual Zeno's half- and half-again,
> embodies for example the usual concept of
> monotone or the constant-progression-of:  time.
>
> Then that for each instant there's a next follows
> from the idea that there exists a time function,
> that the continuously evaluated "next", topologically,
> in the line, exists and is a thing besides that it's not
> except infinitesimally-different from a difference of zero.
>
> So, if you want to be more informed about what the real
> numbers have besides what the ordered field has, and
> consequences of completeness of the real numbers, topologically,
> and for constructive real analysis, infinitesimals are a thing
> and handled their own separate way.  Actually "standard"
> infinitesimals under a definition that works:  models of
> continuous domains like the real numbers include those
> as continuous by line continuity, graphically, by field
> continuity, topologically under the usual convention,
> and by signal continuity, where again effectively establishing
> dense neighborhoods as the topologically.
>
> Here this notion of line continuity and "there are exactly
> infinitely-many infinitesimals uniformly regular through [0,1]",
> can be ignored with usual formal real analysis after algebra
> instead of this "geometric" approach.
>
> But, just because it's ignored, that's not to say that
> "at all scales the numbers aren't uniformly regular",
> because they always are and throughout.
>
>
> And, where it's justified, then in the context that
> must be referring to a particular definition of
> "infinitely-many" and "infinitesimal" that it is so.
> I.e., if something wouldn't make sense, only go
> making sense of it, including making sense that
> "infinity-many" and "infinitesimal" is as simply
> for "n-many" and "n'th", courtesy the bounded and
> piece-wise together all together as the un-bounded.
>
>
> So, introducing "infinity" demands rigor, in mathematics.
>
> And, infinity is already very well introduced to mathematics.
>
>
> If you study or studied calculus you pretty much
> must know that differentials are a refinement of differences,
> as of n-many here not-less-than-infinitely-many equal (constant)
> sized differential regions or patches, as "next" to
> each other as infinitesimals would be.  The region of
> integration, put together of these things all together,
> naturally reflects analyticity.
>
> Then, about the number line, simply consider this:
>    there are points IN the line, each with a next
>      (line continuity, "equivalency function", "time function", "sweep")
>    there are points ON the line, as of limits of sequences that are Cauchy
>      (triangulation, rational and algebraic, ..., complete ordered field)
>    there are points ABOUT the line, as of signal approximation.
>
> Simply disambiguating the language about what differences notions
> of bounds (or ranges) contain values and all the analytical character,
> makes for much more simply making sense of different models of
> real numbers like
>    ..
>    R
>
> and
>    _
>    R
>
> with R-bar and R-dots as each set-theoretic models
> of the continuous domain the real numbers,
> one with line continuity, the other field continuity.
>
> Real-valued functions this way quite well hold up.
>
>
>
> So, "any" "first quantity" "closest to zero" is an
> infinitesimal because it's not a "finite difference"
> that is accessible by a deterministic algorithm.
>
> And, mathematics already has them and the usual thing
> that people know is that the limit from both sides
> establishes meeting in the middle.
>
> I.e., it's a limit of sums and differences besides,
> and no different in either and both.
>
>
> So, please respect that mathematics has thousands of
> years of intuitive and formal infinity and infinitesimals.
>
> Also, please respect that there is a modern foundation
> and besides there are novel retro-classical foundations,
> formalizing and for rigor all sorts of notions of
> mathematical infinities and infinitesimals.
>
>
>
> So, if you want a number line, that is marked with numbers,
> and a first, next, or nearest quantity, when _drawing_ the
> line as if _drawn_ at a steady rate in a straight line,
> there is drawn an entire segment, to draw all of them,
> to draw the first.
>
> This then as simply line-drawing for structure then also
> has simple direct axiomatics, besides as what simplicity
> offers it up as via natural deduction.
>
> In the integer continuum, the first quantity is one.
>
> In the linear continum, with some iota-value, it's one/infinity.
>
> Iota-values as having consecutive differences that sum to one,
> is quite well-defined courtesy exhaustion in the unbounded,
> and "standard" or usual results in the entire formality of
> the integral calculus and real analysis can all be built up in it.
>
>