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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: how
Date: Sat, 13 Apr 2024 09:05:14 -0400
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On 4/13/24 8:35 AM, WM wrote:
> Le 12/04/2024 à 20:57, Jim Burns a écrit :
>> On 4/12/2024 9:18 AM, WM wrote:
> 
>> usedᵂᴹ or not.usedᵂᴹ
>> not.exists
> 
> All individually usable numbers satisfy
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> All collectively useable numbers satisfy
> |ℕ \ {1, 2, 3, ...}| = 0

ALL elements of ℕ are individually usable, there is not finite number n 
that is the highest usable number, which is the fundamental FLAW in your 
logic, which makes it just BLOW UP.

> 
>>> What elements fall between ω and ω*2?
>>
>> ω+1 ω+2 ...
> 
> What elements of {1, 2, 3, ..., ω}*2 fall between ω and ω*2?
> Their distances must be 2.

NONE.

All of the elements of { 1, 2, 3, ...} *2 will be < ω, and the {ω}*2 
part will BE ω*2, and thus not between.

>>
>>> What size has the interval between ℕ*2 and ω*2?
>>
>> I take you to be asking how many ordinals are
>> after each of {2,4,6,...} and before ω+ω
> 
> No, the question is: What elements of {1, 2, 3, ..., ω}*2 fall between ω 
> and ω*2?

None.

>>
>> No doubled ordinal is between ω and ω+ω
> 
> Then {1, 2, 3, ..., ω}*2 increases the gap between {1, 2, 3, ...} and ω 
> from 0 to infinity. Very bumpy. Only a factor 2 is possible when doubling.

Nope, because the "gap" doesn't have a defined distance, just an ordering.

> 
> Regards, WM