Re:How many real numbers exist? New proof moves closer to an...
by magicalist on Thursday July 15, 2021 @12:26PM
(#27846537)
One way to say two sets of things are of equal size is if we can pair up all the elements of one set with all the elements of the other.
If we're ok with extending that to sets of infinite things (we can still pair elements of each set, we'd just never be able to finish listing all the pairs), then we can say that the natural numbers and "the set containing the natural numbers and a sandwich" are of equal size because we could pair 1 from the first set with the sandwich from the second set, pair 2 from the first set with 1 from the second set, 3 from the first set with 2 from the second set, etc etc.
There's no element of either set without a match in the other set, so they have the same cardinality as the natural numbers, with or without the sandwich.
If we're ok with extending that to sets of infinite things (we can still pair elements of each set, we'd just never be able to finish listing all the pairs), then we can say that the natural numbers and "the set containing the natural numbers and a sandwich" are of equal size because we could pair 1 from the first set with the sandwich from the second set, pair 2 from the first set with 1 from the second set, 3 from the first set with 2 from the second set, etc etc.
There's no element of either set without a match in the other set, so they have the same cardinality as the natural numbers, with or without the sandwich.