Re: How many ... (Sparse, Signal, Square Cantor "SPACE")

On 11/02/2024 08:41 PM, Chris M. Thomasson wrote:

On 11/2/2024 3:08 PM, Ross Finlayson wrote:

On 11/02/2024 01:22 PM, Chris M. Thomasson wrote:

On 11/2/2024 11:03 AM, Ross Finlayson wrote:

On 11/02/2024 06:54 AM, FromTheRafters wrote:

WM was thinking very hard :

On 01.11.2024 22:53, FromTheRafters wrote:

WM explained on 11/1/2024 :

On 01.11.2024 19:39, FromTheRafters wrote:

WM formulated the question :

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Infinite subsets don't do that for you, even if you wish really
hard.

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They cannot evade if they are invariable.

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Sets don't change.

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Therefore the elements do not depend on us and our knowledge. "If I
find x, then I can find x + 1" is not relevant. "For every x
(that I
find) there is x + 1" is no relevant. All elements are there,
independent of what we know or do. Therefore the first and the last
are also there independent of us. If they weren't, their existence
would depend on some circumstances and could change.

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Circumstances like "there is no last element"?

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That means, there is always another element. Potential infinity.

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Sets don't change. Forget about amplifying 'not finite' with such as
'actual' and potential' -- infinite simply means not finite and
'actual/potential' is a distinction without a difference. A useless
concept outside of math philosophy.
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the set of denominators have no largest element to 'start' with.

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If all unit fractions are existing, then a smallest unit fraction is
existing. If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions
must
be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of
uncountably many points must be between 0 and x₀. That cannot happen
at x₀ = 0.
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Is that too hard to understand?

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Apparently, for you.

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Au contraire, there are multiple law(s) of large numbers,

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What about this sucker (42^999429994299942) * 2 ?
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Oh, don't forget about ((42^999429994299942) * 2) + 1   damn it!
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>
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and in mathematics like emergence after convergence,
the potential / practical / effective / actual distinction,
of "infinity", is a thing.
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You know who discovered mathematics? Philosophers.
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>

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Well, I frame it about Cantor space, because, there are
some facts deducible from the asymptotic density of zero's
and one's, in the rows and columns Cantor space, even if
there are infinitely or trans-finitely many.

[...]
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Cantor Pairing is fun. Mapping to and fro. Nothing is lost. WM disagrees
here, but shit happens.
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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".