Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Mon, 2 Dec 2024 16:48:34 -0800 Organization: A noiseless patient Spider Lines: 120 Message-ID: References: <4a810760-86a1-44bb-a191-28f70e0b361b@att.net> <23311c1a-1487-4ee4-a822-cd965bd024a0@att.net> <71758f338eb239b7419418f49dfd8177c59d778b@i2pn2.org> <9bcc128b-dea8-4397-9963-45c93d1c14c7@att.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 03 Dec 2024 01:48:34 +0100 (CET) Injection-Info: dont-email.me; posting-host="f777e730b2e8168ebcdb2932c05660c1"; logging-data="3803189"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19jF3XbXXcM/jcnClileI9t4XUSja2Ynes=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:tgApzYastMGlXjWVgdvyZzRysQo= Content-Language: en-US In-Reply-To: Bytes: 5558 On 12/2/2024 4:43 PM, Ross Finlayson wrote: > On 12/02/2024 04:32 PM, Jim Burns wrote: >> On 12/2/2024 9:28 AM, WM wrote: >>> On 02.12.2024 12:53, FromTheRafters wrote: >> >>>> [...] >>> >>> Infinite endsegments contain an infinite set each, >>> infinitely many elements of which >>>  are in the intersection. >> >> Yes to: >> ⎛ regarding finite.cardinals, >> ⎜ for  each  end.segment   E(k) >> ⎜ there is a subset S such that >> ⎝ for each finite cardinal j, j < |S| ≤ |E(k)| >> >> No to: >> ⛔⎛ regarding finite.cardinals, >> ⛔⎜ ⮣ there is a subset S such that ⮧ >> ⛔⎜ ⮤ for  each  end.segment   E(k) ⮠ >> ⛔⎝ for each finite cardinal j, j < |S| ≤ |E(k)| >> >> A quantifier shift tells you (WM) what you (WM) _expect_ >>   but a quantifier shift is untrustworthy. >> >>> An empty intersection cannot come before >>> an empty endsegment has been produced by >>> losing one element at every step. >> >> No. >> Because see below. >> >>> E(1), E(2), E(3), ... >>> and >>> E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ... >>> are identical for every n and in the limit >>> because >>> E(1)∩E(2)∩...∩E(n) = E(n). >> >> They are >> identical COUNTER.EXAMPLES to what you expect. >> >> ---- >>> An empty intersection cannot come before >>> an empty endsegment has been produced by >>> losing one element at every step. >> >> No. >> For the set of finite cardinals, >> EVEN IF NO END.SEGMENT IS EMPTY, >>   the intersection of all end segments is empty. >> >> ⎛ The set of finite.cardinals holds >> ⎜  only finite.cardinals. >> ⎜ >> ⎜ Each finite.cardinal is finite. >> ⎜ >> ⎜ For each finite.cardinal, >> ⎜  only finitely.many finite.cardinals are ≤ it. >> ⎜ >> ⎜ For each finite.cardinal, >> ⎜  only end.segments which start ≤ it >> ⎜ hold it. >> ⎜ >> ⎜ For each finite.cardinal, >> ⎝  only finitely.many end.segments hold it. >> >> ⎛ The set of finite cardinals holds >> ⎜  all finite.cardinals. >> ⎜ >> ⎜ Each finite.cardinal is followed by >> ⎜  another finite.cardinal. >> ⎜ >> ⎜ No finite.cardinal is last. >> ⎜ >> ⎜ The set of finite cardinals has >> ⎜  a subset (itself) which is not two.ended. >> ⎜ >> ⎜ The set of finite cardinals is infinite. >> ⎜ >> ⎜ Each finite.cardinal starts an end.segment. >> ⎜ >> ⎝ There are infinitely.many end.segments. >> >> ⎛ For each finite.cardinal, >> ⎜  only finitely.many end.segments hold it. >> ⎜ >> ⎜ There are infinitely.many end.segments. >> ⎜ >> ⎜ For each finite.cardinal, >> ⎜  not all end.segments hold it. >> ⎜ >> ⎜ For each finite.cardinal, >> ⎜  the intersection doesn't hold it >> ⎜ >> ⎜ EVEN IF NO END.SEGMENT IS EMPTY, >> ⎜⎛ For each finite.cardinal, >> ⎜⎜  only finitely.many end.segments hold it. >> ⎜⎜ For each finite.cardinal, >> ⎜⎜  only finitely.many end.segments hold it. >> ⎜⎜ For each finite.cardinal, >> ⎜⎝  the intersection doesn't hold it >> ⎜ >> ⎜ EVEN IF NO END.SEGMENT IS EMPTY, >> ⎝  the intersection of all end segments is empty. >> >> > > The usual idea of wrestling with a pig is > that you both get dirty, and the pig likes it. > > > Quit letting that pig dirty things. Humm... Indeed. I have heard this before: Never get into an argument with a pig. It will drag you down to its piggish low level and beat you with experience... ;^o