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.
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.