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.
Regards, WM