Re: Does the number of nines increase?

On 7/5/2024 3:53 AM, WM wrote:

Le 04/07/2024 à 20:52, Jim Burns a écrit :

On 7/4/2024 9:49 AM, WM wrote:

Yes, the line point is the limit of an infinite sequence.
Example: 0.999... --> 1.
But the sequence cannot be given other than
by its defining formula like  "0.999...".

>
Givingᵂᴹ the sequence isn't needed in order for us
to know there is only one line.point.

>
Who doubted that?

But it is impossible to
construct or define or find or recognize or communicate
a non-terminating decimal without using a finite formula,
and be it as simple as "0.111...".

Yes, the line point is the limit of an infinite sequence.
Example: 0.999... --> 1.
But the sequence cannot be given
other than by its defining formula like "0.999...".

It is not impossible to construct every desired digit
of the sequence.
But it is impossible to do so without a finite formula.
And it is impossible to desire all digits.
ℵo are always missing.

But uncountability could only be established by completed infinite sequences, not by finite formulas.

The provably.uncountable set of all and only
non.terminating decimals is a subset of 𝒫(N₁×10)
10 = {0,1,2,3,4,5,6,7,8,9}
ℕ₁ = {1,2,3,...}
ℕ₂ = {2,3,4,...} = ℕ₁\{1}
ℕ₁ is well.ordered
Each in ℕ₁ has its first.after in ℕ₂  and
each in ℕ₂ has its last.before in ℕ₁
⎛ ∀S ⊆ ℕ₁:  S={} ∨ ∃j ∈ S: j ≤ᴬ S
⎝ ∀j ∈ ℕ₁: j⁺¹ ∈ ℕ₂  ∧  ∀k ∈ ℕ₂: k⁻¹ ∈ ℕ₁
𝒫(N₁×10) is the set of all sets of
⟨decimal.place,decimal⟩ pairs
A non.terminating.decimal is a set of
⟨decimal.place,decimal⟩ pairs
{⟨1,9⟩} is a set of
⟨decimal.place,decimal⟩ pairs, too,
but it isn't a non.terminating.decimal.
y is a non.terminating.decimal  iff
if, for each decimal.place,
y holds one and only one <decimal.place,decimal>
y:N₁→10  "y is a non.terminating.decimal"
⟺
y ∈ 𝒫(N₁×10)  ∧
∀n ∈ ℕ₁:
∃d ∈ 10: ⟨n,d⟩ ∈ y ∧ ¬∃d₂≠d: ⟨n,d₂⟩ ∈ y
All and only the non.terminating.decimals are in
{y:N₁→10}
{⟨1,9⟩} isn't a non.terminating.decimal and
it isn't in {y:N₁→10}
⎛ But trailing.0s are often assumed, which makes
⎜ {⟨1,9⟩} = 0.9 = 0.9000... = '0.9':N₁→10
⎜ where
⎜ '0.9'{1) = 9
⎜ '0.9'(n) = 0  otherwise
⎜ and '0.9' ∈ {y:N₁→10}
⎜
⎜ In this discussion,
⎜ trailing.0s bring further complication
⎜ without compensating clarity or expressiveness.
⎜ For example, also '0.8999...' ∈ {y:N₁→10}
⎜
⎝ So, here, I'm excluding all trailing.0s decimals.
y:N₁→10:¬∃0… "y is a non.terminating.decimal not.trailing.0s"
⟺
y:N₁→10  ∧
¬∃j ∈ N₁: ∀k>j: y(k)=0
{y:N₁→10:¬∃0…} ⊆ 𝒫(N₁×10)  is the set of
all and only non.terminating.decimals not.trailing.0s

But uncountability could only be established by completed infinite sequences, not by finite formulas.

Uncountability has been established for {y:N₁→10:¬∃0…}
It has been, historically.
It has been, here, for you (WM).
It could be established again, for you,
if you had the slightest interest in being correct,
as distinct from being uncorrected.