Does an order isomorphism between any two disjoint closed intervals of real numbers such as $[0,1]$ and $[3,4]$ exist? If it does, is it unique? Is there any way to explicitly specify it?
I am a beginner so the question is probably pretty dumb, I apologize for that.
Yes, it always exists:$$\begin{array}{ccc}[a,b]&\longrightarrow&[c,d]\\t&\mapsto&c+\dfrac{d-c}{b-a}(t-a).\end{array}$$And, no, it doesn't have to be unique. Take$$\begin{array}{ccc}[0,1]&\longrightarrow&[0,1]\\t&\mapsto&t^2.\end{array}$$