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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Subject: Sparse, Signal, Square, Cantorian "SPACE"
Date: Sun, 3 Nov 2024 09:12:35 -0800
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 >

Here's a way to think about it.


Consider a unit line segment, that's dense rationals.

Then, Cantor "space", starts as the set of all infinite sequences
of 0's and 1's, with a beginning, and no end.

Whether it's a: "space", gets involved as mathematics has a
definition of space, like a "vector space", so as a "set" by
itself, it's more the "image", say, or the "language" in the
formal alphabet {0,1}^w.

So, in "Sparse" Cantor space, is how it's usually considered to
be, that the rationals, are sparse, because the irrationals
are uncountable. "Borel"

Then, the "Signal" Cantor space, has mostly rationals. "Combinatorics"

Then, in "Square" Cantor space, this is that ran(EF) covers
[0,1] and so it's actually where not merely that the rationals
are dense, that ran(EF) is complete.

So, when considering a system of points, dots, on a unit line segment,
and how they relate to the space, then there are at least three
different ways according to "The Law(s) of Large Numbers", whether
they result attenuated and "sparse", emerged and "signal", or
full and "square".


"Borel" and "Combinatorics" and "Line-Drawing" don't "agree" here,
yet each is so, thus according to logic and reason, it's so that
each has some otherwise unstated assumptions, then that each alone
is merely "partial".

Otherwise the formal language of "words" gets into both the
deducibly-sparse and the inductively-constructible-signal,
while line-drawing is fundamentally geometric the "points":
of the actual true Cantor "Space".