Re: More complex numbers than reals?

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

But you claim an empty intersection of
all infinite endsegments,

Each natural number is not.in at least one infiniteⁿᵒᵗᐧᵂᴹ end segments.
Each natural number is not.in the intersection of infinitesⁿᵒᵗᐧᵂᴹ.
Only natural numbers are in natural.number end segments.
The intersection of infinitesⁿᵒᵗᐧᵂᴹ is empty.
Each end segment is infiniteⁿᵒᵗᐧᵂᴹ.

i.e. endsegments which keep
an infinite number of naturals in common with E(1).
That is false.

An infiniteᵂᴹ set is finiteⁿᵒᵗᐧᵂᴹ.but.humongousᴶᴮ.
The finiteⁿᵒᵗᐧᵂᴹ.but.humongousᴶᴮ elements are darkᵂᴹ.
'Humongousᴶᴮ' is intentionally vague, as vague and wobbly as
the distinction between 'darkᵂᴹ' and 'visibleᵂᴹ'.
An infiniteⁿᵒᵗᐧᵂᴹ set is not something you (WM) address.
It is not finiteⁿᵒᵗᐧᵂᴹ, not even finiteⁿᵒᵗᐧᵂᴹ.but.humongousᴶᴮ.
----
ℕⁿᵒᵗᐧᵂᴹ is the set of finiteⁿᵒᵗᐧᵂᴹ.ordinals.
For each finiteⁿᵒᵗᐧᵂᴹ set A
there is a finiteⁿᵒᵗᐧᵂᴹ ordinal j ∈ ℕⁿᵒᵗᐧᵂᴹ such that
{i:i<j} bijects with A
No finiteⁿᵒᵗᐧᵂᴹ ordinal bijects with ℕⁿᵒᵗᐧᵂᴹ
ℕⁿᵒᵗᐧᵂᴹ isn't a finiteⁿᵒᵗᐧᵂᴹ set.
----
Define ordinal j = {i:i<j}
For each finite ordinal j ∈ ℕⁿᵒᵗᐧᵂᴹ
finite successor j⁺¹ = j∪{j} ∈ ℕⁿᵒᵗᐧᵂᴹ
For each finite non.0 ordinal k ∈ ℕⁿᵒᵗᐧᵂᴹ
finite predecessor k⁻¹ = ⋃k ∈ ℕⁿᵒᵗᐧᵂᴹ
For each nonempty subset A ⊆ ℕⁿᵒᵗᐧᵂᴹ
⋂A ∈ A and is first in A
----
Define
f(j) = j⁺¹ = j∪{j}
f: ℕⁿᵒᵗᐧᵂᴹ → ℕⁿᵒᵗᐧᵂᴹ\{0}: 1.to.1
ℕⁿᵒᵗᐧᵂᴹ  ∋  0  ∉  f(ℕⁿᵒᵗᐧᵂᴹ)
'Bye, Bob!
ℕⁿᵒᵗᐧᵂᴹ  is an infiniteⁿᵒᵗᐧᵂᴹ  set.