Re: Does the number of nines increase?

On 6/28/2024 9:50 AM, WM wrote:

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.

>
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.

A set with a cardinal.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
cannot change without its cardinality changing.
Not all cardinals are growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
A set with a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
can change without its cardinality changing.
⎛ ...not that setsⁿᵒᵗᐧᵂᴹ changeⁿᵒᵗᐧᵂᴹ
⎜ Adjunct X∪{Y} isn't X
⎜( Sets can grow by 1 element.
⎜ means
⎝( For each X,Y  X∪{Y} exists
The cardinal ℵ₀ for
all cardinals.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
is a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
| Assume otherwise.
| Assume ℵ₀ is growable.by.1
|
| If
| the cardinal ℵ₀ of
|  all cardinals.growable.by.1
| is growable.by.1
| then
| the cardinal ℵ₀ᐡᴮᵒᵇ for
|  all cardinals.growable.by.1
|  and Bob too
| is growable.by.1 too
| and
| there are more.than.ℵ₀.many
|  cardinals.growable.by.1
|
| However,
| there are ℵ₀.many
|  cardinalities growable.by.1
| Contradiction.
Therefore,
the cardinal ℵ₀ for
all cardinals.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
is a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
Not all cardinals are growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
A set with a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
can change without its cardinality changing.

Therefore cardinality is useless for my proof.

if we stop _saying_
( cardinal of {k……} ̊≤ cardinal of {k⁺¹……}
k ↦ k⁺¹ doesn't stop _being_
1.to.1 from {k……} to {k⁺¹……}
For any difficulty which cardinality presents,
not.saying 'cardinal' not.resolves the difficulty.