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?
Best Answer
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.
Models are generally taken to be sets. The surreal numbers are a proper class: they are "too big" to be considered a non-standard model of the reals in this sense.
But to some extent, we don't really have to insist on models being sets: with suitable set-theoretic axioms, I believe the surreal numbers are also a non-standard model of the reals. In fact, they would be the largest model.
If we pick one particular (set-sized) non-standard model -- e.g. "the" hyperreals -- then we cannot compare its elements to surreal numbers directly. First, we'd have to choose a way to embed the hyperreals into the surreals. There isn't a unique way to do this. In fact, there is a vast number of ways -- an entire proper class of embeddings! (I believe) we can choose to make any particular surreal a hyperreal number by choosing an appropriate embedding.