Re: how

Le 11/04/2024 à 19:36, Jim Burns a écrit :

On 4/11/2024 8:07 AM, WM wrote:

Which ones were unused by e = 2*n?

>
Those of {1, 2, 3, ...} with less than ℵo successors.

 {1,2,3,…} is
the least.upper.bound of all finiteⁿᵒᵗᐧᵂᴹ.{1,2,3,…,k}

Yes, but many natnumbers will never be used.

For each m ∈ {1,2,3,…}:
( for each k ∈ {1,2,3,…}
m+k is a successor of m in {1,2,3,…} )

Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2? What size has the interval between ℕ*2 and ω*2? If you were right, then no elements would fall between ω and ω*2 but all new elements (larger than all in {1,2,3,…}) would stay in {1,2,3,…} while the intervall between ω and ω*2 would be infinite and empty.
Regards, WM