Re: Does the number of nines increase?

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.
[...]

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