Path: ...!weretis.net!feeder6.news.weretis.net!i2pn.org!i2pn2.org!.POSTED!not-for-mail
From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: how
Date: Tue, 23 Apr 2024 22:01:36 -0400
Organization: i2pn2 (i2pn.org)
Message-ID: <v09p60$222fd$4@i2pn2.org>
References: <qHqKnNhkFFpow5Tl3Eiz12-8JEI@jntp> <uvs7k8$1h01f$5@i2pn2.org>
 <5LBLnYAjUgXgTK4Y5LH6e8fCibw@jntp> <uvuuuk$1kece$3@i2pn2.org>
 <wj6ndzp5J9qndM6Ni-9XZdK6jyw@jntp> <v00skm$1m94c$4@i2pn2.org>
 <x_NkVA7tC4PbuDHgWd_lI0r3UuA@jntp> <v012ji$1m94d$4@i2pn2.org>
 <MhMKLiWFUW2BGqeHumLSqLhBo1I@jntp> <v06q93$1uk1v$3@i2pn2.org>
 <Fr-J6IVJwWMeVOUV5nUftUgP93w@jntp>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Wed, 24 Apr 2024 02:01:36 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="2165229"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg";
User-Agent: Mozilla Thunderbird
Content-Language: en-US
X-Spam-Checker-Version: SpamAssassin 4.0.0
In-Reply-To: <Fr-J6IVJwWMeVOUV5nUftUgP93w@jntp>
Bytes: 2462
Lines: 36

On 4/23/24 3:34 PM, WM wrote:
> Le 23/04/2024 à 01:01, Richard Damon a écrit :
>> On 4/22/24 10:15 AM, WM wrote:
> 
>>> The results cannot be compressed to the interval (0, ω) of the set { 
>>> 1, 2, 3, ...}. This shows that new numbers are generated by 
>>> multiplication.
>>>
>> Of course they can be compressed into the interval (0, ω), as every 
>> finite number n < ω, when doubled results in a finite number 2n which 
>> is also < ω.
> 
> Try to map the closed interval [0, ω]*2 = [0, ω*2].
> If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image point 
> in (ω, ω*2]. Infinitely many points remain empty. Crippled mathematics. 
> Ugly. Inacceptable.
> 
> Regards, WM
> 
> 
> 

Why?

[0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural numbers 
(what [0, ω) represents) are closed under multiplication.

The fact that a mixed set of two different classes of ordinals ends up 
with two different classes of ordinals isn't surprizing.

The fact that there are "gaps" in the result isn't surprising, as we see 
the same gaps in the finite part of the set:

0, 1, 2, 3 ... *2 => 0, 2, 4, 6, ...

so the fact that 1 disappeared into a gap in the result says the gap 
between the finites and ω*2 is reasonable.