Re: Replacement of Cardinality

On 8/22/24 8:19 AM, WM wrote:

Le 22/08/2024 à 02:10, Richard Damon a écrit :

On 8/21/24 8:32 AM, WM wrote:

 

No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all and only reciprocals of natural numbers.

 

Can't be, because if it WAS 1/n, then 1/(n+1) would be before it,

 That is tadopted from definable numbers. It is not true for all dark numbers.
 Regards, WM

But you claim the Natural Numbers, which define the whole infinite sequence.
Every Natural Number has, by its definition, a successor, so there is not last one.
And, by your own definitions, if you can use the number individually, which you did for 1/n and thus n, you can use "normal mathematics" on it, that that says that if n exists, so does n+1 as we have a definition for that number, thus there is not last definable number.
Yes, if 1/n was a "dark number" we might not be able to find the n+1 in the dark numbers, but none of those are Natural Numbers, but must be some beyond-finite set of numbers.