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

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

On 17.12.2024 19:52, Jim Burns wrote:

Each finite.cardinal i after 0
is countable.to from.0
∃⟦0,i⦆ smaller than Bobbed ⟦0,i⦆∪{Bob}

>
Stop inventing new nonsense words.

How else does one explain words like 'finite'
to a person who thinks they mean
something which they don't mean?
Ask me what 'Bobbed' means.
⎛ I could have said "Ask me what 'finite' means",
⎝ but you think you know, so you cannot be told.

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.
The infinite end.segments of finite.cardinals
do not include any finite end.segments and
they have an empty intersection.
----
More generally,
for limit.set Lim.⟨A₀,A₁,A₂,…⟩ of
infinite sequence ⟨A₀,A₁,A₂,…⟩ of sets,
x is in Lim.⟨A₀,A₁,A₂,…⟩
(1) if x is in
each of infinitely.many sets in ⟨A₀,A₁,A₂,…⟩
-- with only finitely.many exceptions
and
(2) y is not.in Lim.⟨A₀,A₁,A₂,…⟩
if y is not.in
each of infinitely.many sets in ⟨A₀,A₁,A₂,…⟩
-- with only finitely.many exceptions
⎛ Ask me what 'finite' means.
⎝ It's kinda important.
Not all sequences have limits, because
Not all sequences have, for each potential element,
one of those two conditions holding.
For example, consider the sequence
⟨{0},{1},{0},{1},…⟩
There are more.than.finitely.many exceptions
to 0 being in the sequence and also
more.than.finitely.many exceptions
to 0 not.being in the sequence.
And the same as well for 1
0 and 1 are neither in nor not.in
Lim.⟨{0},{1},{0},{1},…⟩
which is not.allowed for sets.
So, Lim.⟨{0},{1},{0},{1},…⟩ can't be a set.
For a decreasing (i<j ⇒ Bᵢ⊇Bⱼ) sequence ⟨Bₙ⟩
Limⁿ.⟨Bₙ⟩  =  ⋂ⁿ⟨Bₙ⟩
For an increasing (i<j ⇒ Cᵢ⊆Cⱼ) sequence ⟨Cₙ⟩
Limⁿ.⟨Cₙ⟩  =  ⋃ⁿ⟨Cₙ⟩
For sequence ⟨Cₙ⟩
Lim.Infⁿ.⟨Aₙ⟩  =  ⋂ᵐ⋃ᵐᑉⁿ⟨Aₙ⟩
is a lower.bound of the set of
common.with.finite.exceptions elements.
( Lim.Inf.⟨{0},{1},{0},{1},…⟩  =  {}
and
Lim.Supⁿ.⟨Aₙ⟩  =  ⋃ᵐ⋂ᵐᑉⁿ⟨Aₙ⟩
is an upper.bound of the set of
common.with.finite.exceptions elements.
( Lim.Sup.⟨{0},{1},{0},{1},…⟩  =  {0,1}
Assuming Limⁿ.⟨Aₙ⟩ exists,
Lim.Infⁿ.⟨Aₙ⟩  ⊆  Limⁿ.⟨Aₙ⟩  ⊆  Lim.Supⁿ.⟨Aₙ⟩
Assuming Lim.Infⁿ.⟨Aₙ⟩  =  Lim.Supⁿ.⟨Aₙ⟩
Lim.Infⁿ.⟨Aₙ⟩  =  Limⁿ.⟨Aₙ⟩  =  Lim.Supⁿ.⟨Aₙ⟩
and
Limⁿ.⟨Aₙ⟩ exists.