Re: Does the number of nines increase?

On 7/4/2024 4:23 PM, Chris M. Thomasson wrote:

On 7/4/2024 1:01 PM, FromTheRafters wrote:

Providing one infinite set and the other
are both countable or both uncountable.
If one set is size Aleph_zero
and the other is 2^Aleph_zero
then they are not the same size.

>
I was just thinking that infinite is infinite,

Finite is finite, but
infinite is merely not.that.other.thing,
in whatever way it happens to be not.that.
| Happy families are all alike;
| each unhappy family is unhappy in its own way.
|
-- Leo Tolstoy, _Anna Karenina_
Being a thing and not.being that thing
are not symmetric, generally speaking.
More formally,
each set is smaller than its powerset.
Finite or infinite, it's smaller.
ℕ is a smaller infinite than 𝒫(ℕ)
𝒫(ℕ) is a smaller infinite than 𝒫(𝒫(ℕ))
Proof:
There is no way to match
the elements of S to all the subsets of S
Suppose F: S → 𝒫(S) matches
elements of S to subsets of S
Not all the subsets are matched with an element.
In particular, it's a contradiction to claim that
{y ∈ S: y ∉ F(y)} ⊆ S  is matched to any element.
There is no way to match
the elements of S to all the subset of S
S is smaller than 𝒫(S)
If S is infinite,
then S is a smaller infinite than 𝒫(S)