Re: More complex numbers than reals?

On 7/16/2024 3:33 PM, WM wrote:

Le 16/07/2024 à 20:58, Jim Burns a écrit :

On 7/16/2024 9:03 AM, WM wrote:

Le 15/07/2024 à 20:53, Moebius a écrit :

Am 15.07.2024 um 20:31 schrieb Python:

Le 15/07/2024 à 16:46, WM a écrit :

Probably the idea was discussed that
an inclusion-monotonic sequence of
infinite terms
could have an empty intersection.

>
Which is an extremely trivial state of afairs,
Hint:
There is no natural number in
the intersection of all "endsegments".

>
True.

>
And each end segment is infiniteⁿᵒᵗᐧᵂᴹ.

>
Each number that can be used for bijections
has an infinite endsegment.

You (WM) have described ℕᵂᴹ well enough that
it is clear that ℕᵂᴹ isn't ℕⁿᵒᵗᐧᵂᴹ
⎛ Each j in ℕⁿᵒᵗᐧᵂᴹ has in ℕⁿᵒᵗᐧᵂᴹ
⎜ non.0 j⁺¹ immediately.after j
⎜
⎜ Each non.0 k in ℕⁿᵒᵗᐧᵂᴹ has in ℕⁿᵒᵗᐧᵂᴹ
⎜ k⁻¹ immediately.before k
⎜
⎜ Each nonempty subset B ⊆ ℕⁿᵒᵗᐧᵂᴹ holds
⎝ min.B smallest in B.
You (WM) deny some or all of that,
which means
you (WM) aren't referring to ℕⁿᵒᵗᐧᵂᴹ
For ℕⁿᵒᵗᐧᵂᴹ (ℕⁿᵒᵗᐧᵂᴹ is not ℕᵂᴹ),
each end segment is infiniteⁿᵒᵗᐧᵂᴹ.
⎛ The set 𝒟 of
⎜ finiteⁿᵒᵗᐧᵂᴹ end segments of ℕⁿᵒᵗᐧᵂᴹ
⎜ is empty or nonempty.
⎜
⎜ If 𝒟 is empty,
⎜ then
⎜ each end segment is infiniteⁿᵒᵗᐧᵂᴹ.
⎜
⎜ If 𝒟 is nonempty,
⎜ then
⎜ each nonempty E ∈ 𝒟 holds min.E  and
⎜ the set 𝒟ᵐⁱⁿ = {min.E: E ∈ 𝒟}
⎜  is nonempty  and
⎜ exists d = min.𝒟ᵐⁱⁿ ≠ 0 such that
⎜ {j∈ℕⁿᵒᵗᐧᵂᴹ:d≤j} finiteⁿᵒᵗᐧᵂᴹ
⎜ {j∈ℕⁿᵒᵗᐧᵂᴹ:d⁻¹≤j} infiniteⁿᵒᵗᐧᵂᴹ
⎜ {j∈ℕⁿᵒᵗᐧᵂᴹ:d⁻¹≤j} = {j∈ℕⁿᵒᵗᐧᵂᴹ:d≤j}∪{d⁻¹}
⎜
⎜ However,
⎜ (lemma)
⎜ Infiniteⁿᵒᵗᐧᵂᴹ doesn't mean humongous.
⎜ No sets A and {x} exist such that
⎜ A is finiteⁿᵒᵗᐧᵂᴹ and
⎜ A∪{x} is infiniteⁿᵒᵗᐧᵂᴹ.
⎝ Contradiction.
Therefore,
𝒟 is empty,  and
each end segment of ℕⁿᵒᵗᐧᵂᴹ is infiniteⁿᵒᵗᐧᵂᴹ.
⎛ Each element i of ℕⁿᵒᵗᐧᵂᴹ
⎜ is not.in infiniteⁿᵒᵗᐧᵂᴹ {j∈ℕⁿᵒᵗᐧᵂᴹ:i⁺¹≤j}  and
⎜ i is not in all.infiniteⁿᵒᵗᐧᵂᴹ end.segments  and
⎜ i is not in the intersection of all.infinites (all).
⎜
⎝ The intersection of all.infinites (all) is empty.

The ℵo terms of the infinite endsegments
cannot be deleted in steps.

You (WM) are confused about
what an intersection is.

There is no natural number in the intersection of
all infiniteⁿᵒᵗᐧᵂᴹ end segments.

>
What about
the infinitely many numbers which are remainimg from E(1)?

None of them remain in all.infiniteⁿᵒᵗᐧᵂᴹ (all) end segments.

But you claim an empty intersection of
all infinite endsegments,

>
Each natural number is
not.in at least one infiniteⁿᵒᵗᐧᵂᴹ end segments.

>
What numbers constitute
the infinite endsegment which is
common to all infinity endsegments.

All end segments are infiniteⁿᵒᵗᐧᵂᴹ.
∀i ∈ ℕⁿᵒᵗᐧᵂᴹ:
¬∃m ∈ {j∈ℕⁿᵒᵗᐧᵂᴹ:i≤j}:
∀k ∈ {j∈ℕⁿᵒᵗᐧᵂᴹ:i≤j}:
  k≤m
proof.
for k=m⁺¹: ¬(k≤m)
No number is common to
all infiniteⁿᵒᵗᐧᵂᴹ end segments
¬∃m ∈ ℕⁿᵒᵗᐧᵂᴹ:
∀i ∈ ℕⁿᵒᵗᐧᵂᴹ:
  m ∈ {j∈ℕⁿᵒᵗᐧᵂᴹ:i≤j}
proof.
for i=m⁺¹: ¬(m ∈ {j∈ℕⁿᵒᵗᐧᵂᴹ:i≤j})
An infiniteⁿᵒᵗᐧᵂᴹ end segment
common to all infiniteⁿᵒᵗᐧᵂᴹ end segments
not.exists.