Post by Michael MoroneyPost by Mitch RaemschZero 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.