Re: Does the number of nines increase? (size, measure, number)

On 07/05/2024 10:25 PM, Ross Finlayson wrote:

On 07/05/2024 07:12 PM, Moebius wrote:

Am 06.07.2024 um 03:58 schrieb Moebius:

Am 05.07.2024 um 20:08 schrieb Jim Burns:
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The other way around.
It's set.inclusion which stops working as a guide to size.

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Right. How would we be able to compare, say, the sets {1, 2, 3, ...}
and {-1, -2, -3, ...} concerning "size" by relying on "set inclusion"?
Or, say, {1, 2, 3, ...} and {1.5, 2.5, 3.5, ...} etc.
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Or even {1, 2, orange} and {1, 2, 3}.

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Or let's compare the size of, say,
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{0, 1, 2, 3, ...} with the size of {(0, x_0), (1, x_1), (2, x_2), (3,
x_3), ...} (for some x_0, x_1, x_2, x_3...).
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It seems that in this case the size of these two sets should be the
same, I'd say.
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Now let's compare the size of, say, {1, 2, 3, ...} with the size of
{(y_1, 1), (y_2, 2), (y_3, 3), ...} (for some y_1, y_2, y_3, ...).
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Again, it seems that in this case the size of these two sets should be
the same, I'd say.
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So what's the size of the set {(0, 1), (1, 2), (2, 3), (3, 4), ...}?
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The same as the size of {0, 1, 2, 3, ...} and/or the same as the size of
{1, 2, 3, ...}?
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The "conclusion" seems to be that {0, 1, 2, 3, ...} and {1, 2, 3, ...}
have the same size, even though {1, 2, 3, ...} c {0, 1, 2, 3, ...}.

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I'm reminded many years ago, when studying size relations in sets,
that one rule that arrived was that a proper subset, had a size
relation, smaller than the superset, and was told that it was
not so, while, still it was written how it was so.
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Then, Fred Katz pointed me to his Ph.D. from M.I.T. and OUTPACING,
showing that it was a formal result that it was so.
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So, the "conclusion", seems to be, "not a conclusion",
for all the "considerations", their conclusions, together.
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Then another one was asymptotic density and the size relation
of sets not just being ordered but also having a rational value,
this was the "half of the integers are even".
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It involves a bit of book-keeping, yet, it is possible to
keep these various notions, while still there's cardinality
sort of in the middle, where of course on the other side
of these refinements of the notion of the relation of size
in infinite sets of numbers their spaces their elements,
then there's an entire absolute of "ubiquitous ordinals",
that have the infinite sets as of a "size".
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So, when you mean cardinal, say cardinal. There
are other notions of "size", and "measure", and, "number".
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Set theory is pretty great, and cardinals are natural
in ordinary set theory, set theory: a study of a theory
of objects with one relation "elt", in the ordinary,
with no infinite descending epsilon-chains (well-foundedness).
Extra-ordinary set theory is even greater,
the fuller theory of objects as elements
and relation.