Given two totally ordered sets, suppose that each of them can be embedded in the other one; if this is the case, are they mutually isomorphic?

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Antonio Pasini

The answer to my question is negative. Here is an example. Let F be any bijection from the real line R to [0,1] and define an ordering

I also realize that my question was ill stated: I forgot to say that the two ordered sets to consider where supposed to be defined over the same set of elements, otherwise R and [0,1] immediately provide a counterexample. Nevertheless, that additional condition doesn't change things so much.