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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: how
Date: Wed, 24 Apr 2024 13:07:36 -0700
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On 4/24/2024 10:16 AM, WM wrote:
> Le 24/04/2024 à 04:01, Richard Damon a écrit :
>> 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.
> 
>> Why?
> 
> Continuity.
>>
>> [0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural numbers 
>> (what [0, ω) represents) are closed under multiplication.
> 
> Impossible if (0, ω) is completely filled. Every 2n is larger than n.
[...]

How can it be 100% completely filled when its unbounded?