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

On 12/18/2024 7:40 AM, WM wrote:

On 18.12.2024 02:16, Jim Burns wrote:

On 12/17/2024 5:12 PM, WM wrote:

We have the sequence of
intersections of endsegments
f(k) = ∩{E(1), E(2), ..., E(k)}
with E(1) = ℕ
and the definition of that function
∀k ∈ ℕ :
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k}
and the fact that
∩{E(1), E(2), ...} is empty.
More is not required to prove
the existence of finite endsegments.

>
Here is a counter.example to your claimed requirement:
There are no finite.cardinals common to
all the infinite end.segments of finite.cardinals.

⎛ Sets of sheep, sets of pebbles, etc are
⎜ smaller.than fuller.by.one sets.
⎜
⎜ fuller.by.one B∪{x}: B ⊉ {x}
⎜
⎜ B smaller.than fuller.by.one sets  ⇔
⎜ B ⊉ {x}  ⇒  #B < #(B∪{x})
⎜
⎜ Some other sets are
⎜ other.than smaller.than fuller.by.one sets.
⎜ They are NOT.smaller.than fuller.by.one sets.
⎜
⎜ Y NOT.smaller.than fuller.by.one sets  ⇔
⎝ Y ⊉ {x)  ⇒  #Y = #(Y∪{x})

There are no definable cardinals
common to all endsegments of definable cardinals.

For each set smaller.than fuller.by.one sets,
fuller.by.ONE sets are smaller.than
fuller.by.TWO sets.
⎛ #B < #(B∪{x})  ⇒
⎝ #(B∪{x}) < #(B∪{x,x′})
There is no set which is largest among
the sets smaller.than fuller.by.one sets.
⟦0,ℵ₀⦆ is the set of cardinals of
  sets smaller.than fuller.by.one sets
(of sheep, of pebbles, ...).
⎛ #B < #(B∪{x})  ⇒
⎜ ∃k ∈ ⟦0,ℵ₀⦆:  #⟦0,k⦆ = #B
⎜
⎝ ¬∃j ∈ ⟦0,ℵ₀⦆:  ¬(#⟦0,j⦆ < #⟦0,j+1⦆)
There is no cardinal largest among cardinals of
  sets smaller.than fuller.by.one sets
⎛ ∀j ∈ ⟦0,ℵ₀⦆:
⎜ #⟦0,j⦆ < #⟦0,j+1⦆
⎜ #⟦0,j+1⦆ < #⟦0,j+2⦆
⎜ ∃k ∈ ⟦0,ℵ₀⦆:  k = #⟦0,j+1⦆ > j
⎜
⎜ ¬∃j ∈ ⟦0,ℵ₀⦆:
⎝ ∀k ∈ ⟦0,ℵ₀⦆:  j ≥ k

There are no definable cardinals
common to all endsegments of definable cardinals.

There are no cardinals of
  sets smaller.than fuller.by.one sets
common to all end.segments of cardinals of
  sets smaller.than fuller.by.one sets.
⎛ ¬∃j ∈ ⟦0,ℵ₀⦆:
⎜ ∀k ∈ ⟦0,ℵ₀⦆:  j ≥ k
⎜
⎜ ¬∃j ∈ ⟦0,ℵ₀⦆:
⎝ ∀k ∈ ⟦0,ℵ₀⦆: j ∈ ⟦k,ℵ₀⦆

Finite cardinals are in all non-empty endsegments.

Only cardinals of
  sets smaller.than fuller.by.one sets
are in non.empty end.segments of cardinals of
  sets smaller.than fuller.by.one sets.
Q. What does 'finite' mean?

The infinite end.segments of finite.cardinals
do not include any finite end.segments and
they have an empty intersection.

>
Explicitly wrong.
As long as only infinite endsegments are concerned their intersection is infinite.

Cardinal k of a set smaller.than fuller.by.one sets
#⟦0,k⦆  =  #B  <  #(B∪{x})
Set ⟦0,ℵ₀⦆ of each.and.only cardinals of
sets smaller.than.fuller.by.one sets
End.segment ⟦k+1,ℵ₀⦆ of cardinals of
  sets smaller.than fuller.by.one sets
contains subset ⟦k+1,k+j+2⟧ larger.than ⟦0,j⦆
and is an end.segment which k is not in common with.
Each cardinal of
  sets smaller.than fuller.by.one sets
is not.in.common.with
an end.segment of cardinals of
  sets smaller.than fuller.by.one sets,
which end.segment is larger than each cardinal of
  sets smaller.than fuller.by.one sets
⎛ ∀k ∈ ⟦0,ℵ₀⦆:
⎝ k ∉ ⟦k+1,ℵ₀⦆  ∧  #⟦k+1,ℵ₀⦆ >ᵉᵃᶜʰ ⟦0,ℵ₀⦆

----
More generally,
for limit.set Lim.⟨A₀,A₁,A₂,…⟩ of

>
There are no limits involved.

Then it would be great if you (WM)
stopped calling things 'limit sets'
Q. What is a limit set?