Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail 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 References: <9aac1c05-e35c-4b94-97cd-2b66a6877ca8@att.net> <4-mdnRvc88GjSBX7nZ2dnZfqn_idnZ2d@giganews.com> From: Ross Finlayson Date: Sat, 6 Jul 2024 12:44:32 -0700 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: <4-mdnRvc88GjSBX7nZ2dnZfqn_idnZ2d@giganews.com> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 7bit Message-ID: Lines: 75 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-5adwiPAoE1TvSoAthD0sxrzuoyGtqw6evddyKjVTtR/mQYd1gzBwrOm1ZOPQUYtgiJn2KDQ8L2YLTtP!o5wwZphp+QxekwAROGxD78elwos2gSag1pXEgY6X6IEWZm7m562XgwWeEhQ1F7v1jvhjNGCiMUY1 X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 4531 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.