Re: Incompleteness of Cantor's enumeration of the rational numbers (cardinals)

On 11/17/2024 11:36 AM, Ross Finlayson wrote:

On 11/17/2024 10:25 AM, Ross Finlayson wrote:

On 11/17/2024 10:16 AM, Ross Finlayson wrote:

On 11/17/2024 08:59 AM, FromTheRafters wrote:

WM pretended :

On 17.11.2024 12:38, FromTheRafters wrote:

WM presented the following explanation :

On 17.11.2024 12:01, FromTheRafters wrote:

WM was thinking very hard :

On 16.11.2024 22:33, Moebius wrote:
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For example "aleph_0 - aleph_0" is not defined.

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Small wonder. ℵo means only infinitely many: |ℕ|, |ℚ|, and many
others.
|ℕ|-|ℕ| however is defined.

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No, it is not.

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If sets are invariable then ℕ \ ℕ is empty.
If |ℕ| concerns only the elements of ℕ, then |ℕ|-|ℕ|= 0.

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So, you're saying that if I take aleph_zero natural numbers and I
remove the aleph_zero odd numbers from consideration in a new set, I
will have a new emptyset instead of E?

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Try to understand. "aleph_0 - aleph_0" is not defined.

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Try to understand that |N| equals aleph_zero.

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I'm wouldn't say from experience that it'll ever learn.
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There's no such thing as subtraction in cardinals,
only ^ and <.
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There's setminus w.r.t. sets: that though results what are that
sets have cardinals, being in an equivalence class
of Cartesian bijections with all other members of
the same cardinal an equivalence class as after the
Cantor-Schroeder-Bernstein theorem and Cartesian bijections
that making the equivalence relation, ..., sometimes
people confuse that with initial ordinals yet that's
considered wrong as ordinals and cardinals are different,
and a cardinal is an equivalence class of sets, though
it gets rather involved how big they are, when "1" the
cardinal is the largest equivalence class in ZF.
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Then of course according to me, there are some
non-Cartesian functions like the natural/unit equivalency function,
or "EF", "sweep", not-a-real-function yet standardly modeled by
real functions, so that it can demonstrate for the analog/digital
and continuous/discrete a Jordan measure/content a line segment
geometrically as a continuous bounded domain, countable.
This of course is all framed up about "anti-diagonal" the
Cartesian and "only-diagonal" the non-Cartesian, since
the "anti-diagonal" sometimes sort of wrongly called
"The Diagonal Argument" is used for so many things
in ordinary algebras about set theory with cardinals.
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It sort of works out that this "EF" is "unique"
this way, with regards to "exists" and "exists-unique",
yet there's also the, "Reverse EF", which is pretty
much the same with either choice of endpoints.
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That's just about "defining line-drawing" when one
actually puts a paper to pencil and marks a mark,
an opus, a fait, a work, a line segment, point A
to point B, the shortest, straightest line,
here for simplicity between zero and one, and
BECAUSE simplicity between zero and one. (For infinity.)
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In multiple dimensions then this makes a
"spiral space-filling curve", like "scribbling a dot".
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Which is a bit more accessible than trans-finities,
though, they're all part of the same theory.
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Like, "fill in the oval", ..., though without
crossing lines, that line-drawing is abstract
Platonic perfect straight lines, and, scribbling
a dot is an abstract perfect ever-widening dot,
that results an abstract perfect Platonic circle.
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Have you ever bothered to wonder that "1",
the cardinal, the equivalence class of singletons,
has the "top", cardinal?
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This is where any larger set has all-that-many
singletons, and, singletons is the largest
equivalency class, of individuals, in all the theory.
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Then, one might even wonder that "1's cardinal",
sees that there are more finites than in-finites,
in a sort of summary, of "super-cardinals".
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There's that "large cardinals" are neither sets
nor cardinals, yet, if cardinals are in set theory,
aren't all cardinals sets? Or, perhaps cardinals
are independent the set theory, otherwise.
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Well, now you have.
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"One: is the loneliest number, ...."
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Mers, M'b'ers, Nu'm'b'ers, ....
This is a brief etymological account of Mer
the Atlantean, M'ba, then Nu'm'ba, Nubian,
N'um'b'mers, where the word "numbers" comes from.
Traditionally the story of counting arrived
from a woman at a fire, tossing stones in _time_,
in ancient, ancient Africa, establishing both
_counting_ and _counting time_, that much,
much later is a shepherd stashing pebbles
for his sheep in a bag.