The definition of an ordered set according to W. Rudin in his book, Principles of Mathematical Analysis is:
An ordered set is a set S in which an order is defined
He also defined order in his book:
Let S be a set. An order on S is a relation, denoted by <, with the following two properties:
- If x, y ∈ S then one and only one of x < y, x = y, x > y is true.
- If x, y, z ∈ S and x < y and y < z then x < z.
I can't think of any set that doesn't have an order. Is there any set that is not ordered?
A counterexample would be very helpful.
Best Answer
A set isn't ordered unless you supply an ordering. A set on its own is not an ordered set.
So the natural numbers is not an ordered set. The natural numbers with the standard $<$ is an ordered set. Usually, when in the context of ordered sets, just saying "the natural numbers" will implicitly imply "with the standard ordering". But if we're being pedantic and strict about it, then it needs to be said explicitly; if it isn't we don't have an ordered set.
And, of course, there are plenty of sets without a standard total ordering. Like the complex numbers. Or the points on a sphere. Or the integers modulo $10$. Many of these sets can be given a total order, and in many cases it is even pretty easy to do so. But it won't play nicely with the standard structures those sets do have, so that's of limited use. And there is no canonical way of doing it, so if you want others to know which order you're talking about you have to say exactly what your order is.
And then there is the possibility of sets which can't be given a total order at all. It's impossible to mention concrete examples here, because no set can be proven un-orderable, at least in ZF.