Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Thu, 13 Mar 2025 15:56:05 +0100
Organization: A noiseless patient Spider
Lines: 38
Message-ID: <vqurm5$3brpp$2@dont-email.me>
References: <vqrbtd$1chb7$2@solani.org> <vqrn89$u9t$1@news.muc.de>
 <vqrp47$2gl70$1@dont-email.me> <vqrtn3$1uq5$1@news.muc.de>
 <vqs1og$2k7oh$2@dont-email.me> <vqsh1r$2cnf$1@news.muc.de>
 <vqsoq5$2p6pb$1@dont-email.me> <vqsuf0$2g64$1@news.muc.de>
 <vqt0hv$2r2iu$1@dont-email.me> <vquqsj$1vla$1@news.muc.de>
Reply-To: invalid@example.invalid
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Thu, 13 Mar 2025 15:56:06 +0100 (CET)
Injection-Info: dont-email.me; posting-host="6e446472212fd7ec6fec2aadb76cb810";
	logging-data="3534649"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+aAW5xGXCjoTStwsh1C+fT"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:XUjzouab2zDiDKp0p7v9SHTQN2s=
In-Reply-To: <vquqsj$1vla$1@news.muc.de>
Content-Language: de-DE
Bytes: 2341

Am 13.03.2025 um 15:42 schrieb Alan Mackenzie:
> Moebius <invalid@example.invalid> wrote:

>> Should read: |ℕ \ {1}| = ℵo
> 
> Ah.  Thanks!  If only people could write what they meant.

People? Mückenheims? :-P

>> Should read: and [for all n e ℕ:] if |ℕ \ {1, 2, 3, ..., n}| = ℵo then |ℕ \ {1, 2, 3,
>> ..., n+1}| = ℵo.
> 
> OK.
As you mentioned, "\" (usually) is used for _sets_, but not for cardinal 
numbers.

On the other hand, concerning the latter, we might write:

|ℕ| - |{1}| = ℵo  (Since |ℕ| = ℵo, |{1}| = 1 and ℵo - 1 = ℵo.)

and for all n e ℕ:

if |ℕ| - |{1, 2, 3, ..., n}| = ℵo then |ℕ| - |{1, 2, 3, ..., n+1}| = ℵo .

(Since |ℕ| = ℵo, for all n e ℕ: |{1, 2, 3, ..., n}| = n and for all n e 
ℕ: ℵo - n = ℵo.)

Of course, his "if ... then ..." claim is "nonsense", since simply

      for all n e ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo

as well as

      for all n e ℕ: |ℕ| - |{1, 2, 3, ..., n}| = ℵo .

No idea, what he is striving towards.