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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Does the number of nines increase?
Date: Fri, 05 Jul 2024 10:33:50 -0400
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Peter Fairbrother was thinking very hard :
> On 05/07/2024 10:59, FromTheRafters wrote:
>> After serious thinking Peter Fairbrother wrote :
>>> On 04/07/2024 21:01, FromTheRafters wrote:
>>>> Chris M. Thomasson laid this down on his screen :
> [..]
>>>> 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 imagined the 9's as being the digits of 0.99... so distinguishable and 
>>> countably infinite.
>> 
>> There is nothing wrong with a sequence having duplicate members. Sets (ZFC) 
>> don't have duplicates though.
>
> Oh dear. Let's forget about the nines (which are not duplicates, the nth 9 is 
> different from the (n+1)th 9).
>
> One set is all the natural numbers, the second is that plus an orange. OK? 
> Both countable infinities, no duplicates.

Yes, this is exactly the infinite cardinal arithmetic WM doesn't grasp. 
Or rather, say, he objects to its validity.

> [...]
>
>> A difference between the idea of 'same' and 'equal size'. 'Same' if each 
>> can be a subset/superset of the other (matching elements) and 'equal size' 
>> in terms of cardinality (pairing elements).
>
> Maybe. but I'm not concerned about "same" here. only size.

In that case you need not have any concern about what the elements are, 
only cardinality.

> And the elements 
> are pairable (if an orange can be paired to a number), but one set has an 
> orange in it and the other doesn't.

That doesn't matter, it's another element even if it's a fish.

> So, pairing (and cardinality) don't really work for sizing these sets; in 
> everyday terms, the second is bigger.

Pairing does work, I suppose a better term would be equinumerosity.

> Certainly heavier.:)

If elephants were elements.