Re: Incompleteness of Cantor's enumeration of the rational numbers

On 27.11.2024 19:02, joes wrote:

Am Wed, 27 Nov 2024 18:13:06 +0100 schrieb WM:

Decreasing every element in the real interval (0, 1] by one point
yields the real interval [0, 1). The set of points remains the same, the
set of positive points decreases by 1.

What is "decreasing by a point"?

That can only be performed for half-open intervals because most points are dark.

 

Replacing every element of the set {0, 1, 2, 3, ...} by its successor
yields {1, 2, 3, ..., ω}. The number of ordinals remains the same, the
number of finite ordinals decreases.

There is no natural whose successor is omega.

If there is nothing between ℕ and ω, then replacing every element of the set {0, 1, 2, 3, ...} by its successor yields {1, 2, 3, ..., ω}.
Regards, WM