# What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain:

• real numbers
• a hierarchy of infinitesimal numbers like $\epsilon, \epsilon^2, \epsilon^3, \ldots$
• a hierarchy of transfinite numbers like $\omega, \omega^2, \omega^3, \ldots$ where $\omega = 1/\epsilon$

and both allow the four standard arithmetic operations to be applied to any combination of real, infinitesimal, and transfinite numbers. So what is the difference, if any, between these number systems? If the Wikipedia statement is accurate, what numbers are surreal but not hyperreal?

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There are many non-isomorphic non-standard models of reals; any of them can be called hyperreals, although one specific model (the ultrafilter construction on $\mathbb{R}^\mathbb{N}$) is often called "the" hyperreals.
For any (set-sized) non-standard model of the reals, call it $F$, we can construct a new non-standard model of the reals extending $F$ that contains an $F$-infinitessimal $\epsilon$: that is, $\epsilon$ is a positive number that is smaller than every positive element of $F$. Therefore, $F$ can't be dense in this extension: e.g. in this new model, the interval $[3+\epsilon, 3+2\epsilon]$ doesn't contain any elements of $F$. – Hurkyl Oct 27 '12 at 8:25