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?