Re: Does the number of nines increase?

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.

Sets can have elements inserted,
which makes them different sets.
The effect on size of inserting 1
is not the same for all sets.
Larger than all the
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
are the
infiniteⁿᵒᵗᐧᵂᴹ.sizes.which.cannot.grow.by.1
For each
finiteⁿᵒᵗᐧᵂᴹ.size.which.can.grow.by.1
in the set ℕ of all
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
there is a larger
finiteⁿᵒᵗᐧᵂᴹ.size.which.can.grow.by.1
It would lead to contradictions for
for the size of the set ℕ of all
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
to be smaller than one of the
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
_in itself_
So,
the size of the set ℕ of all
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
isn't any of the
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
Only
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
can grow by 1,  which
the size of the set ℕ of all
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
isn't any of.
The size of the set ℕ of all
finiteⁿᵒᵗᐧᵂᴹ.sizes.which.can.grow.by.1
cannot grow by 1
It is an
infiniteⁿᵒᵗᐧᵂᴹ.size.which.cannot.grow.by.1
The effect on size of inserting 1
is not the same for all sets.