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NNTP-Posting-Date: Sat, 06 Jul 2024 19:44:23 +0000
Subject: Re: Does the number of nines increase? (size, measure, number)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 6 Jul 2024 12:44:32 -0700
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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:
>>>
>>>> The other way around.
>>>> It's set.inclusion which stops working as a guide to size.
>>>
>>> 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.
>>>
>>> Or even {1, 2, orange} and {1, 2, 3}.
>>
>> Or let's compare the size of, say,
>>
>> {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...).
>>
>> It seems that in this case the size of these two sets should be the
>> same, I'd say.
>>
>> 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, ...).
>>
>> Again, it seems that in this case the size of these two sets should be
>> the same, I'd say.
>>
>> So what's the size of the set {(0, 1), (1, 2), (2, 3), (3, 4), ...}?
>>
>> The same as the size of {0, 1, 2, 3, ...} and/or the same as the size of
>> {1, 2, 3, ...}?
>>
>> 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, ...}.
>
> 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.
>
> 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.
>
> So, the "conclusion", seems to be, "not a conclusion",
> for all the "considerations", their conclusions, together.
>
> 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".
>
> 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".
>
> So, when you mean cardinal, say cardinal. There
> are other notions of "size", and "measure", and, "number".
>
>
>

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.