Re: Does the number of nines increase?

On 7/14/2024 9:16 AM, WM wrote:

Le 13/07/2024 à 23:55, Jim Burns a écrit :

On 7/13/2024 11:43 AM, WM wrote:

Le 12/07/2024 à 22:23, Jim Burns a écrit :

On 7/12/2024 1:04 PM, WM wrote:

says a person who believes that
by exchanging two elements
one of them can be lost.

>
I have never claimed ℵ₀ = 2

>
You have claimed that
by exchanging X and O an O can disappear,

>
Yes.

>
That disqualifies you from any serious discussion.

E pur si muove.
f(j)  :=  (j=Bob ? 1 : j⁺¹)
(Perl ternary conditional operator)
f: ℕ₁⁺ᴮᵒᵇ → ℕ₁: 1.to.1
Bob ∉ f(ℕ₁⁺ᴮᵒᵇ)

in fact infinitely many can disappear
by pure exchange.

>
Yes.

>
Laughable.
Are you aware that nobody followed you?

E pur si muove.
g⟨j,k⟩  :=  ⟨ (j+k-1)(j+k-2)/2+j, 1 ⟩
g: ℕ₁×ℕ₁ → ℕ₁×{1}: 1.to.1
∀⟨j,k⟩ ∈ ℕ₁×(ℕ₁\{1}):  ⟨j,k⟩ ∉ g(ℕ₁×ℕ₁)

What we also [understand which] you (WM) don't is that
ℕ₁⁺ᴮᵒᵇ and ℕ₁×ℕ₁ aren't finiteⁿᵒᵗᐧᵂᴹ sets,

>
Logic remains valid for all correct mathematics,
finite and infinite.

Finiteⁿᵒᵗᐧᵂᴹ.set A
can be finitelyᴶᴮ.ordered.
A finiteᴶᴮ.order '<' of set A
is trichotomous, and
each nonempty.subset B of A
holds a firstᑉ B.element and a lastᑉ B.element.
An infiniteᴶᴮ.order '<₂'of set C
is trichotomous and not finiteᴶᴮ --
one or more nonempty.subset D of C
_lacks_ a firstᑉ² D.element or a lastᑉ² D.element.
No set has
both a finiteᴶᴮ.order and an infiniteᴶᴮ.order.
The familiar order '<' of familiar ℕ
is an infiniteᴶᴮ.order.
Familiar ℕ is an infiniteⁿᵒᵗᐧᵂᴹ set.
Each nonempty subset B of ℕ
in the familiar order '<'
holds a firstᑉ B.element.
For each j in ℕ
there is nextᑉ j⁺¹ > j in ℕ\{0}
For each k in ℕ\{0}
there is nextᑉ k⁻¹ < k in ℕ
E pur si muove.