(April 7, 2019 at 1:49 pm)Smaug Wrote:(April 6, 2019 at 10:46 pm)Jehanne Wrote: An actual infinity means a set (say, the natural numbers) has a cardinality that is infinite, as opposed to a "potential infinite", which is always finite, even given the fact such "grows" forever.
Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?
It's a subtle distinction. Virtually all mathematicians would say that the set of prime numbers is an actual infinite, whose cardinality is identical to the set of natural numbers; most philosophers would agree with this, also. However, the set of real numbers (both rational and irrational) is, as Cantor proved, an infinite set whose cardinality is greater than the set of natural numbers, even though both are infinite sets. And, so, some infinities are bigger than others. The set of natural numbers (and, prime numbers, as well as rational numbers) is a countably infinite set whereas the set of real numbers is uncountable.
