Re: Does the number of nines increase?

On 6/25/2024 4:18 PM, WM wrote:

Let the infinite sequence 0.999...
be multiplied by 10.
Does the number of nines grow?

Cardinalities which can grow by 1 are finite.
The number of nines in 0.999... is
larger than each finite cardinality.
It does not equal any finite cardinality.
It cannot grow by 1
tl;dr
No.

Corollary-question:
Does the number of nines grow
when in 0.999 the decimal point is shifted
by one or more position?

By only one or any finite number of positions,
no.
Nuance:
There are _only_ positions in 0.999... which
are separated by some finite number,
even though there are infinitely.many of them.
The positions in 0.999... correspond to
numbers in well.ordered inductive ℕᴬ⤾⁺¹₀ᐣ⤓
for each j ∈ ℕᴬ⤾⁺¹₀ᐣ⤓
⟨0…j⟩ ̊< ⟨0…j⁺¹⟩ ̊≤ ℕ ̊= ℵ₀
⟨j……⟩ ̊≤ ⟨j⁺¹……⟩ ̊≤ ⟨j……⟩ ̊= ℕ ̊= ℵ₀
tl;dr
No.