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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Does the number of nines increase?
Date: Thu, 4 Jul 2024 21:44:09 -0700
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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.
> 
> β„• is a smaller infinite than 𝒫(β„•)
> 𝒫(β„•) is a smaller infinite than 𝒫(𝒫(β„•))
> 
> Proof:
> There is no way to match
> the elements of S to all the subsets of S
> 
> Suppose F: S β†’ 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 βˆ‰ F(y)} βŠ† SΒ  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