Re: Does the number of nines increase?

Am Fri, 28 Jun 2024 18:19:11 +0000 schrieb WM:

Le 28/06/2024 à 19:16, joes a écrit :

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}?

It is |ℕ| - 1. That was easy.

Why is that the same as ℕ\{1}? They are not supersets of each other.

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?

Number of elements.

Which you define how? Starting maybe with |ℕ|. What numbers do you use?

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.

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

Yes, it is |ℕ|.

Which is as a concrete number…? [not OT?]

The effect of removing a nine from 0.999... is changing its value.

Only when replacing with another digit.

When in 0.999... the decimal point is shifted, the number of nines
remains constant. That has nothing to do with cardinalities.

Yes it does; that is how WE define the number of nines.

You can’t remove from the right, since there is no end.

If all are a complete set, then we can shift all by one position to the
left.

Or equivalently, add 9.
It is as a rope coming from the machine: there’s always more.

You can only remove from the left (dividing by 10)

Removing from the right is done by multiplying by 10.

Where do the superfluous digits go: 0.(9)9 ?

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Am Fri, 28 Jun 2024 16:52:17 -0500 schrieb olcott:
Objectively I am a genius.