Re: Does the number of nines increase?

Le 27/06/2024 à 18:30, Jim Burns a écrit :

On 6/27/2024 8:03 AM, WM wrote:

Cardinalities which can grow by 1 are finite.
Cardinalities which cannot grow by 1 are infinite.

Cardinalities are useless.
Sets can grow by 1 element.
The set ℕ of indices of the nines in 0.999... cannot grow by a natural number.

Try to think.
"Countable" is an unsharp notion,

 Consider instead the notion '1.to.1 function'

That is as unsharp. It shows equinumerosity of primes and rationals.

 Because ℕ is not larger than ℕ
the cardinality of ℕ cannot grow by 1

but
there are sets which are
ℕ with 1 element inserted or deleted.
Those sets have the same cardinality as ℕ
The cardinality didn't grow by 1

As I said cardinality is useless to check equinumerosity.

The set of numbers and of indices is ℕ.

Therefore in 0,999... there are precisely as many nines as in 9.999....
Therefore behind the decimal point there are less after shifting it.
Regards, WM