Thread: Linear order isomorphism

Hello,

I have a linear order relation defined on the product of set of complex numbers



I can prove that this relation is a linear order that's dense and without endpoints. But I have a problem with this question:

Is this order isomorphic with ? is a normal greater or equal relation on the set of real numbers.

For order to be isomorphic there has to exist a bijection between two sets that preserves the relation between the elements, meaning that if there's a bijection then for some if . Do I have to create a bijection , because I have trouble coming up with one, or is there a simpler way to answer this question?

Interesting. The theory of dense linear orders without endpoints is categorical only for ω and not for uncountable cardinals. So, 〈ℂ,R〉and 〈ℝ,≤〉can be non-isomorphic. In fact, they are. For a hint, see this Dr. Math question.