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

On 1/7/2025 4:13 AM, WM wrote:

On 06.01.2025 23:43, Jim Burns wrote:

On 1/5/2025 1:14 PM, WM wrote:

ℕ cannot be covered by FISONs,
neither by many nor by their union.
If ℕ could be covered by FISONs
then one would be sufficient.

>
ℕ is the set of finite.ordinals.
ℕ holds each finite ordinal.
ℕ holds only finite.ordinals.

A finite.ordinal k = ⟦0,k⦆ is an ordinal which
is smaller.than fuller.by.one sets,
that is, k for which #⟦0,k⦆ < #⟦0,k+1⦆
ℕ = ⦃finite⦄ is the set of ordinals which
are smaller.than fuller.by.one sets.

|ℕ| is by definition
the smallest transfinite number,

|ℕ| = #ℕ is by definition
the smallest transfinite cardinal, that is,
the smallest cardinal #Y of any set Y which
is NOT smaller.than.fuller.by.one sets.
ω = ⟦0,ω⦆ is by definition
is NOT smaller.than fuller.by.one sets (transfinite) and
IS the first among the (well.ordered) ordinals which are
NOT smaller.than.fuller.by.one sets (the transfinites).
¬(#⟦0,ω⦆ < #⟦0,ω+1⦆)
¬(#⟦0,ξ⦆ < #⟦0,ξ+1⦆)  ⇒  ω ≤ ξ
∀ᵒʳᵈj,k < ω:
#⟦0,j⦆ = #⟦0,k⦆  ⇔  ⟦0,j⦆ = ⟦0,k⦆
#⟦0,ω⦆ = #⟦0,ω+1⦆  ∧  ⟦0,ω⦆ ≠ ⟦0,ω+1⦆
The cardinal:ordinal distinction
-- which does not matter in the finite domain
matters in the infinite domain.