Re: Does the number of nines increase?

After serious thinking Peter Fairbrother wrote :

On 05/07/2024 15:33, FromTheRafters wrote:

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.

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I imagined the 9's as being the digits of 0.99... so distinguishable and countably infinite.

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There is nothing wrong with a sequence having duplicate members. Sets (ZFC) don't have duplicates though.

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Oh dear. Let's forget about the nines (which are not duplicates, the nth 9 is different from the (n+1)th 9).
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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.
 

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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).

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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,

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Eh? I don't follow that. AFAICT every element in one set can be paired with an element of the other, so if that is a definition of being the same size, they are the same size.

That is it, and for finite cases cardinality (size) works out as 'how many elements' but for transfinite cases 'same size' (equinumerosity) is more appropriate.

But one set contains an orange more than the other one does, so afaict pairing doesn't work here.

It still does, but is a bit non-intuitive.