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I am trying to understand the extensions of the real numbers. Nothing too serious like a course or research. Especially since the wikipedia article on superreal numbers is pretty empty. (https://en.wikipedia.org/wiki/Superreal_number).

Are the surreal numbers just the union of reals, hyperreal, and superreal numbers?

I am sorry if this question is too vague; I do not need a super rigorous explanation. I am looking for a more "naive" (or intuitive) explanation.

The Bosco
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  • Wikipedia says "The field of superreals is itself a subfield of the surreal numbers." From that phrase it sounds like the superreals are strictly smaller than the surreals. And that makes sense, as there are too many surreals for them to be contained in any set (any ordinal is a surreal, for instance), while the superreal numbers are contained in a set. I wrote about the ordinals and the surreals in an answer to another question here – Arthur May 03 '18 at 06:49
  • My problem is more with understanding what a superreal number is. Why are they different than hyperreals for example? – The Bosco May 03 '18 at 06:51

1 Answers1

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Unfortunately this Wikipedia page is misleading. The relevant notion is that of super-real fields, introduced by Dales and Woodin in their monograph Super-real fields.

Super-real fields are up to isomorphism fraction fields of quotients by prime ideals of algebras of continuous real valued functions on completely regular spaces. They include what they call hyper-real fields, which are quotients of the same algebras by maximal ideals, which themselves include ultrapowers of $\mathbb{R}$ for some ultrafilter on a cardinal. Dales and Woodin proved that those fields are real closed and gave conditions on the prime ideals for the unique underlying field order to be an $\eta_1$-set.

In informal terms, super-real fields form a class of ordered field extensions of $\mathbb{R}$, and may be small or big. The surreal numbers of birthdate $<\kappa$ for some cardinals $\kappa$ are super-real and even hyper-real fields.

Thus the phrasing "superreal numbers" is bad, just like it would be bad phrasing to say that a "ring number" is an element of a ring. Each super-real field embeds in the field of surreals, but this is nothing specific to super-real fields.

There is a nice short presentation of those fields and their purpose which you can find here.

nombre
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