Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?



Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?
There is no such countably infinite set, but there is such a set with size $2^{\aleph\_0}$. These are classic results of Dushnik & Miller [[*Concerning similarity transformations of linearly ordered sets*](http://www.ams.org/journals/bull/1940-46-04/S0002-9904-1940-07213-1/home.html), Bull. Amer. Math. Soc. 46 (1940), 322-326]. I also outlined the idea behind this construction in my [first MathOverflow answer](https://mathoverflow.net/questions/9901/which-are-the-rigid-suborders-of-the-real-line/9932#9932)!


