Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 08.01.2025 20:19, Jim Burns wrote:

On 1/8/2025 4:16 AM, WM wrote:

On 08.01.2025 00:50, Jim Burns wrote:

 

The cardinal:ordinal distinction -- which does not matter in the
finite domain matters in the infinite domain.

>
The reason is that the infinite cardinal ℵ₀ is based on the mapping of
the potentially infinite collection of natural numbers n,
all of which have infinitely many successors.
The cardinal ℵ₀ is not based on the mapping of the actually infinite
set ℕ where ℕ \ {1, 2, 3, ...} = { }.

 
For each set smaller.than a fuller.by.one set, the cardinal:ordinal
distinction doesn't matter.
Cardinals and ordinals always go together.
For each set smaller.than a fuller.by.one set there is an ordinal of
its size in the set ℕ of all finite ordinals.
Each set for which there is NOT an ordinal of its size in the set ℕ of
all finite ordinals is NOT a set smaller.than a fuller.by.one set.

 
The set {1, 2, 3, ...} is smaller by one element than the set {0, 1, 2,
3, ...}. Proof: {0, 1, 2, 3, ...} \ {1, 2, 3, ...} = {0}. Cardinality
cannot describe this difference because it covers only mappings of
elements which have almost all elements as successors.