Re: Replacement of Cardinality

On 8/2/24 10:58 AM, WM wrote:

Le 02/08/2024 à 01:45, Richard Damon a écrit :

On 8/1/24 8:02 AM, WM wrote:

 

What is immediately before ω? Is it a blasphemy to ask such questions?

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It has no predicessor, just like in the Natural Numbers 0 has nothing before it.

 0 has a continuum above it, no gap! Likewise there must be no gap below ω.

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You can expand your number system to define number there, which seems to be what you "dark numbers" are, numbers bigger than all the finite Natural Numbers, but smaller than w.

 Thank you.
 

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It is not a contradiction to my formula if some n has no n+1.

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It is a violation of the DEFINITION of the Natural Numbers.

 Otherwise there is a contradiction of mathematics: separated unit fractions.
 Regards, WM
 

No, all Unit fractions are separated in distance by a finite amount, it is just that you can't specify any finite distance that separates all unit fractions.
The reversal of the order of qualifications causes the problem.
This is the nature of unbounded numbers, something your logic doesn't seem to be able to handle.
Every number 1/n is separated from the next smaller unit fraction, 1/(n+1) by a distance of 1/(n*(n+1)) which is a value that is greater than zero, so we always have a finite difference between all unit fractions, but that distance gets arbitrarily small, so we can't choose a single finite eps that all unit fractions are seperated by, even though all unit fractions are seperated by a finite distance.
This just shows that this spacing, APPROACHES 0, as a limit, as n increases. But approaching a limit of 0 is not the same as being 0.