Now, we have decided a point (I here introduce Tverse's spelling: "piont"; see "neologisms") must be the very "first thing."
(Here's a mild pretty little graphic of a "Piont" for your enjoyment, or use your own imagination.)
How many are there?
The Origin Sequence:
First, there wasn't even "nothing"; there was only this piont.
It is the only one.
Or is it?
It is the absolute first thing; there was not any thing "before" it, so whatever it can be, it must be and, as it is dimensionless, it
can be an infinite number of these without there being the slightest way to tell which it is, "one" or "many."
Therefore, Paradox: (link; earlier version, this ideation)
It must be only one, yet it must also be "an infinite number."
Paradox-resolution: It must somehow be, or become, both:
"An infinite number" implies individuality. Yet as
"one," no individuals, thus, something must happen:
They move, creating dimensionality.
All at once and every one from every other, they move.
Instantly there is "volume," in the shape of an hollow sphere.
(If to begin with there is only one "location," and if every one of the infinite number of pionts
moves away from every other, it should be seen this manufactures a sphere-shape.
This is what was meant earlier in my website's history by the never-completed html file
"The Notion of a Sphere").
In this spherical shell. "one point thick," pionts are still trying to get far, far away from other pionts. Thus the sphere is expanding.
Every piont has two properties, obtained from the "creation" of them all:
1) Existence. "Is-ness." 'not-[No Thing]-ness'. (go back to our discussion of "[No Thing]" for this one's justification...)
2) Dynamic individuality.(we just established this).
This individuality is absolute; it is as if each piont sought to be the only piont in existence.
We might think of each piont as being surrounded by an infinitely extended, perfectly grainless region, strongest right next to the piont, weakening with distance from it.
of repulsion toward all other pionts
" dMin "; Minimum Distance; The minimum allowable distance between pionts
(Please read "The Magus and His Notylon Spheres," for
a story-form illustration of one of the Paradoxes involved in this explanation....
You should peruse also the Euler link, and the Web of course, for more on why these pionts have arranged themselves in/on the surface of the hollow sphere the way that they do arrange themslves (see "Euler wakes and stomps around a bit")
It should now be seen that these points have separated, and that thus, between any two close-together pionts is some distance.
Were this distance not present, the two pionts would be ONE piont, violating one of Tverse's origin principles, so ever and always there must be some distance between pionts.
What I call "minimum distance" is defined as follows:
"That distance, less than which, two pionts would become one piont."
It might be expressed, using math symbols, in this silly little way:
" 0+ "
Or, you may glance at this graphic....
Now we have both a "separation axiom" (or, at worst, a "separation definition"), and with it, the possibility of a METRIC.
But we do not have a measurement with which to relate this possibility-of-a-metric to some "size" level in standard Physics, nor are we likely to easily get one.
Asking for a measurement amount —asking "how long is this dMin, this "minimum distance"?— is asking the wrong question.
Since this is absolutely the smallest imaginable distance, we should not ask "how big is it in relation to us?," but instead must ask "How big are WE in relation to IT ?"
NOTE ON THE REMAINDER OF THIS FILE:
What follows was in the original file, but is not necessarily pertinent to the "separation axiom."
I have left it in this new file just in case it might help understanding. You may read on, or not, as you choose.
(If not, backbutton out of here to the TVerse index, please)
Pionts are now as individualized as they can get, or are they?
"Volume" has appeared; "volume" is three
Euclidean dimensions, but the sphere-shell is Euclidean two-dimensional.
Pionts can now further individualize, by flooding from 2D
Implosion of pionts into 3D volume (interior of the sphere-shell)
occurs, filling it up, more piling in, Endless, Forever.
Remaining: The One: this[globe]remains
as always, forver will, resisting expansion while spewing new
into 'volume', into interior,inwardly, like crazy.
Remaining: The Many: An infinite
set of dimensionless, noncoalesceable pionts,
less than half inside the Tverse-Universe interior, the rest
still in 2D, in Tverse-Universe boundary,
pouring 'out' into interior and :
Remaining: The Compression: The
One spews pionts; interior existing pionts puch back,
a Network of primarily Vertex Order Twelve
forms in the interior.
Remaining, newly born: Operational
Tetrahedraverse, and the possibility for Physics.
Consider a stretched rubber sphere, say, fifty feet across,
filled with a very large number of miniscule frictionless
balls. Consider further that these balls are so tiny they cannot
be seen. Consider that, if the sphere were to be filled with the
balls while it was "loose," then let go so as to
compress the balls together "from all directions" ('spherical
compression'), these balls would naturally arrange themselves
into a network of vertex order twelve.
This network is a situation in which every ball has twelve ---and
only twelve-- immediate neighbors; it is, of course, the
"problem of the thirteen spheres" in every place within
the compressing sphere (except, naturally, the outermost layer of
balls, which have only six neighbors).
Please try to visualize this, and recall the Magus and his
incompressible, noncoalesceable, frictionless spheres, which in
the infinite extent of smallness become dimensionless.
They, while being dimensionless, yet may not become ONE.
This is the very meaning of "noncoalesceable," and
some thinking about this should provide you with the conclusion
that even if dimensionless, the points in Tetrahedraverse
may not fall back into, or by any power whatsoever be forced back
into on another.
An infinite number of dimensionless,
noncoalesceable points must constitute an
infinitely large collective 'object.'
Enjoy thinking about that, please. It is rather critical to
your understanding of Tetrahedraverse.
Back to the Index Page....