Is there a one-to-one correspondance between the real numbers and the hyperreal numbers?
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It depends on what you mean by ${}^\mathbb R$. There are structures of different sizes, any of which is a reasonable version of ${}^\mathbb R$, there is no nice uniqueness theorem here as with the reals. Maybe you mean something more concrete, with ${}^*\mathbb R$ being an ultrapower of $\mathbb R$ by a non-principal ultrafilter on $\mathbb N$? – Andrés E. Caicedo Apr 27 '16 at 00:08