Posted: Sep 14, 2017 9:33 am
Well, that is not quite how completeness is defined in set theory, but roughly yes.Wortfish wrote:Newmark wrote:
That is indeed correct. Infinite sets doesn't need to include every possible element. The negative integers don't include all the integers, the integers don't include all the reals, the reals don't include all the complex numbers, etc. It is also quite possible to have an set that excludes specific elements, e.g. Z \ {0} (all the integers except 0), which is still a set of (countably) infinite size. More importantly, it is entirely mathematically possible for an infinite set to have an upper and/or lower bound, e.g. the negative numbers.
Not sure about that. If an infinite set does not include every possible element, it is not complete.
However, a infinite set does not have to be complete ... or include "every possible element", for that matter.
For instance, the set of rational numbers is not complete, but it is still countable infinite. There is not a finite count of real numbers that can be exactly represented as the ratio of two integers, but neither can, as one of so many examples, √2 be exactly represented as the ratio of two integers.
(Yes, Wortfish, you can approximate it; but you have to approximate it, because you cannot be exact.)