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

On 27.11.2024 13:32, Richard Damon wrote:

On 11/27/24 5:12 AM, WM wrote:

Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1 is consistent.

 So you think, but that is because you brain has been exploded by the contradiction.
 We can get to your second set two ways, and the set itself can't know which.
 We could have built the set by the operation of removing 1 like your math implies, or we can get to it by the operation of increasing each element by its successor, which must have the same number of elements,

Yes, the same number of elements, but not the same number of natural numbers.
Hint: 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.
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.
Regards, WM