Re: Does the number of nines increase?

Am Fri, 28 Jun 2024 13:50:06 +0000 schrieb WM:

Le 28/06/2024 à 06:31, Jim Burns a écrit :

On 6/27/2024 2:37 PM, WM wrote:

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

 

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.

So what is the size of N\{2}?

Sets can have elements inserted, which makes them different sets.
The effect on size of inserting 1 is not the same for all sets.

It is changing the infinite set but not its cardinality. Therefore
cardinality is useless for my proof.

What are you replacing it with?

The size of the set ℕ of all finite.sizes.which.can.grow.by.1
cannot grow by 1 It is an infinite.size.which.cannot.grow.by.1

Your discussion is off topic.

The size of N, containing all finite numbers, is itself infinite.

The effect on size of inserting 1 is not the same for all sets.

The effect of removing a nine from 0.999... is changing its value.
When in 0.999... the decimal point is shifted, the number of nines
remains constant. That has nothing to do with cardinalities.

You can’t remove from the right, since there is no end. You can only
remove from the left (dividing by 10), or cut it off, replacing all
following places with zeroes. If you removed one from the middle (i.e.
not the first one), there are still infinitely many.