Re: Does the number of nines increase?

On 6/29/2024 1:18 PM, WM wrote:

Le 29/06/2024 à 15:24, Jim Burns a écrit :

On 6/28/2024 9:50 AM, WM wrote:

It is changing the infinite set
but not its cardinality.

>
A set with a cardinal.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
cannot change without its cardinality changing.

>
Cardinality is irrelevant.
>

Therefore cardinality is useless for my proof.

>

For any difficulty which cardinality presents,
not.saying 'cardinal' not.resolves the difficulty.

>
Cardinality does not present a difficulty.
It is simply unable to distinguish |ℕ| and |ℕ_0|.

A not.at.all.drug.impaired use of the word 'irrelevant'
declaring that cardinality is irrelevantᵂᴹ to
cardinals |ℕ₁| and |ℕ₀|
Cardinality does not distinguish cardinals |ℕ₁| and |ℕ₀|
because
j ↦ j⁺¹ is 1.to.1 from ℕ₀ to ℕ₁
and
k ↤ k is 1.to.1 to ℕ₀ from ℕ₁
Without 'cardinality'
you (WM) still have your (WM's) difficulty,
that
j ↦ j⁺¹ is 1.to.1 from ℕ₀ to ℕ₁
and
k ↤ k is 1.to.1 to ℕ₀ from ℕ₁

All nines of 0.999... are from
the sequence 0.9, 0.09, 0.009, ...
None of the ℵo nines makes
its partial sum 0,9, 0.99, 0.999, ... equal to 1.

Each non.empty.set of 9s holds a first.in.set 9
Each 9 has a first.after 9 and a last.before 9,
except the first 9, which only has a first.after 9
None of those is the last 9
None of those equal 1

ℵo nines fail to make 0.999... = 1.

1 is near almost.all (all.but.finitely.many) of
  0.9  0.99  0.999  0.9999  0.99999  ...
for any sense > 0 of 'near'
For that reason,
we assign 1 to 0.999...
The values of infinite.length decimals
are assigned by a different method from how
the values of finite.length decimals are assigned.