Re: Does the number of nines increase?

Am 05.07.2024 um 16:59 schrieb Peter Fairbrother:

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

Peter Fairbrother was thinking very hard :

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.
>

So, pairing (and cardinality) don't really work for sizing these sets;

Sure it does.

in everyday terms, the second is bigger.

If you say so. But we are dealing with math terms here (you know).

Pairing does work,

Right.

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.

Right.

But one set contains an orange more than the other one does,*)

Right. But how does this orange contribute to the "size" of the (whole) set in "set theoretic terms" (i.e. power/cardinality).
It does not "count" (sic!). (lol)
The set {orange, 0, 1, 2, 3, ...} is /countably infinite/, just like {0, 1 2, 3, ...}.
Proof: There's a bijektion between {orange, 0, 1, 2, 3, ...} and {0, 1 2, 3, ...}, say, 0|-> orange, 1 |-> 0, 2 |-> 1, etc.
This means: |{orange, 0, 1, 2, 3, ...}| = |{0, 1, 2, 3, ...}| or card({orange, 0, 1, 2, 3, ...}) = card({0, 1, 2, 3, ...}).
Sure: {0, 1, 2, 3, ...} c {orange, 0, 1, 2, 3, ...}. But this does not imply that {orange, 0, 1, 2, 3, ...} is "larger" from a set-theoretic point of view.
Maybe helpful:
You might think that {orange, 1, 2, 3, ...} is "larger" than, say, {1, 2, 3, ...}, right?
On the other hand, you might agree that {orange, 1, 2, 3, ...} has "the same size" as, say, {{orange}, {1}, {2}, {3}, ...}, right?
But what if I tell you (now) that in our special context we had defined
1 = {orange}, 2 = {1}, 3 = {2}, etc.
Then actually, {{orange}, {1}, {2}, {3}, ...} = {1, 2, 3, 4, ...}.
So {orange, 1, 2, 3, ...} would have the same size as {{orange}, {1}, {2}, {3}, ...}, but on the other hand it would be larger... :-P
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*) "But one set contains an orange more than the other one does." Right, {orange, 0, 1, 2, 3, ...} contains 1 orange, and {0, 1, 2, 3, ...} contains 0 oranges. Hence {orange, 0, 1, 2, 3, ...} contains exactly 1 orange more than {0, 1, 2, 3, ...}. (Note that we are talking about finitely many oranges here.)