Re: Does the number of nines increase?

On 6/26/2024 3:15 AM, WM wrote:

Le 26/06/2024 à 00:11, Jim Burns a écrit :

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.

>
Is the set of natural indices complete
such that
no natural number can be added?

 You'd do better at being understood if
you said what distinguishes 'natural number'
from 'natural index'
if you draw a distinction,
if being understood is something you want.
  The well.ordered inductive natural.number.indices
are complete
such that
no natural.number.index is not
a natural.number.index.
 The well.ordered inductive natural.number.indices
are complete
such that
no natural.number.index is without
its natural.number.index.successor
and
such that
no nonempty natural.number.index.set is without
its natural.number.index.first.member.
 The well.ordered inductive natural.number.indices
are complete
such that
no natural.number.index.pair is without
its natural.number.index.sum.
 

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⤓

>
And they are fixed.

 They are certainly well.ordered and inductive.
 If, by 'fixed', you mean that
each is not anything other than itself,
then yes.
They are fixedly well.ordered and inductive.
 

Therefore your answer is correct:No.

 

Therefore
9.999... has one 9 less [one 9 fewer]
after the decimal point than 0.999... .

 No.
"Move" the decimal point.
0.9⃒9999...  and  09.9⃒9999...
0.99⃒999...  and  09.99⃒999...
0.999⃒99...  and  09.999⃒99...
0.9999⃒9...  and  09.9999⃒9...
0.99999⃒...  and  09.99999⃒...
...
 Inductive:
for each 9  there is its successor.9
There isn't one 9 fewer.

 Therefore,
once again,
'humongous' and 'infinite' are different.