Re: how

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 :

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What is a gap in the ordinals?
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It is a not existing ordinal between two ordinals.

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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.

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Ja, solche natürlichen Zahlen gibt es in der Tat nicht.

 Below ω there is a huge gap or there are dark ordinals.

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 > the idea that ω follows upon all natural numbers with no gap in between.
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Hint: There is no "gap in between" in the following sense:
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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.