Re: More complex numbers than reals?

On 7/18/2024 9:09 AM, WM wrote:

Le 18/07/2024 à 13:18, Jim Burns a écrit :

On 7/17/2024 1:57 PM, WM wrote:

What remains to keep the endsegments infinite?

>
| For each number, there is a number after
 proves false
| There is a number after all numbers.

>
The claim is that
after every n defining the endsegment E(n)
there are ℵo natnumbers remaining from E(1)
in all infinite endsegments.

ℵ₀.many remain in E(n)
0.many remain in all.infinite (all) end segments
n is in E(n) and not.in E(n+1) and not in.all
n+1 is in E(n) and not.in E(n+2) and not in.all
n+2 is in E(n) and not.in E(n+3) and not in.all
n+3 is in E(n) and not.in E(n+4) and not in.all
...
The claim is that
⎛ Each j in E(n) has in E(n)
⎜ non.n j⁺¹ immediately.after j
⎜
⎜ Each non.n k in E(n) has in E(n)
⎜ k⁻¹ immediately.before k
⎜
⎜ Each nonempty subset B ⊆ E(n) holds
⎝ min.B smallest in B.
It follows not.first.false.ly from that claim
and the claim about ℕ that
⎛ Each j in ℕ has in ℕ
⎜ non.0 j⁺¹ immediately.after j
⎜
⎜ Each non.0 k in ℕ has in ℕ
⎜ k⁻¹ immediately.before k
⎜
⎜ Each nonempty subset B ⊆ ℕ holds
⎝ min.B smallest in B.
that
for fₙ(j) = j+n
fₙ: ℕ → E(n): 1.to.1
|ℕ| ≤ |E(n)|
Plus
ℕ ⊇ E(n)
|ℕ| ≥ |E(n)|
|ℕ| = |E(n)|
|ℕ|.many remain in E(n)
For each n ∈ ℕ
fₙ(j) = j+n is 1.to.1  and
|ℕ|.many remain in E(n)
Infiniteⁿᵒᵗᐧᵂᴹ does not mean humongous.

You (WM) aren't referring to ℕⁿᵒᵗᐧᵂᴹ

>
I am referring to mathematics.

But you aren't referring to mathematicsⁿᵒᵗᐧᵂᴹ.

Most of my "peers" are too stupid to comprehend this,
not all, fortunately, and no students.

Thus, not mathematicsⁿᵒᵗᐧᵂᴹ.

The sequence of endsegments is inclusion monotonic.

The sequence of
  step.up non.0.step.down well.order ℕ
end.segments is inclusion.monotonic.

Every infinite endsegment has
an infinite intersection with all infinite endsegments.

No number in
  step.up non.0.step.down well.order ℕ
is in every
  step.up non.0.step.down well.order ℕ
end.segment.
'Intersection' means that
no number in
  step.up non.0.step.down well.order ℕ
is in the intersection of all
  step.up non.0.step.down well.order ℕ
end.segments.
'For each, exists' does not imply 'exists, for each'
'Infinite' does not mean 'humongous'.