Re: Does the number of nines increase?

On 7/4/2024 9:35 PM, Jim Burns wrote:

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)
 

Does saying that the reals are infinitely denser than the naturals make any sense? As in there are more reals than naturals even if they both are infinite? This can mess with me sometimes! Damn it. ;^o