Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 19.10.2024 16:08, FromTheRafters wrote:

WM expressed precisely :

On 19.10.2024 13:15, FromTheRafters wrote:

WM was thinking very hard :

On 19.10.2024 12:56, Jim Burns wrote:

>

What is immediately before ω

No number exists immediately before ω

>
Immediately or not: The number before ω is finite.

>
No number is before omega,

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In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, emerging, produced, i.e., as we put it, with the potential infinite. [Hilbert].

 Is this from a 1925 lecture?#

Yes. [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]

 

All these becoming, emerging, produced numbers

 What does that even mean?

It means that you cannot use, name, identify all natural numbers. There is always a greatest one for you. This is either the last one existing or the last one before the dark realm.

omega is the naturally ordered set of natural numbers.

>
That is another meaning of the word but not relevant, in particular not if there is no set of natural numbers because new numbers  can be produced and added.

 ???

If we multiply all natural numbers by 2, then the number of numbers does not change but the interval is doubled. That means the result contains larger numbers than we have multiplied. If these are natural numbers too, then we have not multiplied all.
If we process all natural numbers and get more than we have processed, then this is not a set but a potentially infinite collection, because sets do not change.
Regards, WM