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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: how
Date: Mon, 22 Apr 2024 18:58:57 -0400
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On 4/22/24 11:10 AM, WM wrote:
> Le 21/04/2024 à 18:07, Moebius a écrit :
>> Am 20.04.2024 um 20:34 schrieb WM:
>>> Le 20/04/2024 à 19:42, FromTheRafters a écrit :
>>
>>>> What is a gap in the ordinals?
>>>>
>>> It is a not existing ordinal between two ordinals.
>>
>> Aha, dann gibt es also zwischen allen Ordinalzahlen "gaps".
> 
> More precisely: A gap on the ordinal axis is a not existing ordinal in 
> distance 1 from an existing ordinal. Example: Between -oo and 0 there us 
> a huge gap.
> 
>>> It is a not existing natural number next to ω, for instance.
>>
>> Ja, solche natürlichen Zahlen gibt es in der Tat nicht.
> 
> Below ω there is a huge gap or there are dark ordinals.
>>
>>  > the idea that ω follows upon all natural numbers with no gap in 
>> between.
>>
>> Hint: There is no "gap in between" in the following sense:
>>
>> ω is (provably) the smallest ordinal AFTER (large than) all finite 
>> ordinals. 
> 
> If so, then at ω-n there are gaps.
> 
> Regards, WM
> 
> 

Yes, there is sort of a gap below ω as you can only get to ω via a 
"hyper" step, not a normal step, like from 1 to 2.

There are no Ordinals in that gap, as the set below it is unbounded, but 
in a sense, there is a gap, since you can back down from ω, or step up 
to ω with ordinary steps, only "hyper" steps, which go from finite, to ω 
to 2*w, to 3*ω and so on.