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  1. I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers?
  2. I know that the order type of rationals $\eta$ is a homogeneous order type (meaning that for any two its elements there is an automorphism that sends one to another). Are the order types of irrationals and real transcendental numbers homogeneous as well?
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    For both the rationals and the real algebraics, there is a way of viewing inclusion into $\mathbb{R}$ as giving a "Dedekind-completion". Let $\varphi:\mathbb{Q} \to \mathbb{A}$ (where $\mathbb{A}$ denotes the real algebraics) be an order isomorphism. A suitable uniqueness result for Dedekind completions then implies there is a unique order isomorphism $\varphi^* \ : \mathbb{R} \to \mathbb{R}$ extending $\varphi$. This map $\varphi^*$ then restricts to an order isomorphism between $\mathbb{R} \setminus \mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{A}$.2012-02-07

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