An uncountable infinity minus a countable infinity is an uncountable infinity

When you say "plus," you implicitly mean the union of two sets, right?

I.e. "A countable infinity plus a countable infinity" means "The cardinality of two sets that are both countable"...?

I think the diagnalisation proof works just as well.

The real numbers are uncountable (cannot be listed).
The natural numbers are countable (can be listed) and a subset of the real numbers.

If the real numbers without the natural numbers were countable they would be listable.

List them. You now have a list of all the real numbers that aren't natural numbers.

Construct a new number that is different from the first item in the first place, different to the second item in the second place, different to the third item in the third place, etc. When constructing, do not use 0.

You now have a number that is not a natural number, but is not in your list. Therefore you cannot create a complete list, therefore the uncountable set of reals minus the countable set of naturals results in an uncountable set.

Sethrates: uh, sure, I guess, though I can't think of any other definition of "plus" or "minus" such that this would be sensical and false.

T Kybernetikos, picking an example does not prove the rule. You need to choose generally.

Well, any countably infinite set can be mapped onto the natural numbers, so it sort of is a general proof. Not every uncountably infinite set can be mapped onto the real numbers, but I think (anyone care to correct me) that all the ones that can't be, can't be because they're bigger (and so, would still not result in a countable set). If someone would like to tell me a set that is smaller than the reals but still uncountable, I'd be glad to learn.

hm. i think the rationals are smaller and i can't recall whether or not it is countable. i think it is, though.

You mean, (aleph-one) or greater minus (aleph-null) is never ? <Taken from aleph number>

Why naturally. ;D

Al: a wholly accurate pun.