I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ''numbers''.

And, if I well understand, we have a chain of inclusions as: $Reals \subset Hyperreals \subset Superreals \subset Surreals$ but, really, I don't well understand the difference between them.

So, before taking on a more detailed study, I have some ''naive'' question:

What kinds of elements are in one of these extensions but not in the other?

It is possible to have more ''great'' extensions or the chain stops with the Surreals?

These extensions have some utility in other branches of mathematic that justify the effort of study them?

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    $\begingroup$ Someone pointed me to this article a couple of days ago, which is a sort of exploration of what is restrictive enough to define the real numbers. It's not directly related, but some of the examples have proofs using the extensions of the reals, which might give you a better feel for some of them. (And also has the best discussion of $0.9999\dotsm = 1$ that I've ever seen.) $\endgroup$ – Chappers Oct 7 '15 at 20:36
  • $\begingroup$ Re: "These extensions have some utility in other branches of mathematic that justify the effort of study them?" - that's not what justifies the study of a mathematical object in my opinion. $\endgroup$ – Noah Schweber Feb 14 '18 at 15:53

First of all, what you call "Reals, Hyperreals, Superreals and Surreals" are different type of objects. The real numbers form a unique field $\mathbb{R}$ (up to isomorphism), hyperreal fields and superreal fields form classes of non-isomorphic fields, and the surreal numbers form a unique Field $No$ (up to isomorphism), meaning it is a proper class equipped with operations and an order that are also proper classes (informally, things too big to be sets).

Secondly, $\mathbb{R}$, $No$ and all hyperreal fields and superreal fields are real closed fields: fields with the same first order properties (first order sentences in the language of rings) as the field of algebraïc real numbers. And all these fields can be ordered in a unique way. The class of real closed fields is a very rich one, it has model-theoritic properties, geometric properties, set-theoritic properties, of course algebraïc properties, and an interesting - and still mysterious - spectrum of topological properties.

Thirdly, important properties of those fields are not given by the order type of their set infinitesimal elements, and one can't necessarly identify two ordered fields because they have isomorphic (let alone informally comparable) structures of infinitesimals. I would even say focusing on their types infinitesimals is a bad way to "understand" them. I could tell you that there are infinitesimals such as $(1,\frac{1}{2},\frac{1}{3},...)$ in some hyperreal fields, as $\frac{1}{\omega_2}$ in the field of surreal numbers but I don't think it would do much more than entertaining you.

-The ordered field of real numbers is unique up to isomorphism as a field that satisfies the Least Upper Bound property. It is of course the field where the most part of analysis takes place, and the most part of continuous mathematics takes place in structures whose definitions involve $\mathbb{R}$ at some crucial point. It is also the field used by physicists.

$\mathbb{R}$ is archimedean, meaning every real number is less than some natural integer, and it is up to isomorphism the unique archimedean ordered field with no proper dense extension, the unique Cauchy-complete archimedean ordered field, the unique archimedean ordered field satisfying the Nested Interval Property. (I think you can find these facts and much more in Real Analysis in Reverse)

Quite importantly, every archimedean ordered field embeds in $\mathbb{R}$ in a unique way.

-Hyperreal fields are proper extensions of $\mathbb{R}$. As such, they are not archimedean, so there are elements in any hyperreal field greater than every positive integer or lower than every negative integers. Those elements are called infinite elements and their multiplicative inverses are called infinitesimals. They also have an important property that for any countable or finite subsets $A,B$ such that $A < B$, there is $x$ in the hyperreal field wuch that $A < x$ and $x < B$. Hyperreal fields are interesting because of this property, and some of them can be used in Non Standard Analysis or in model theory. You can find the definition of a hyperreal field on Wikipedia. I suggest you look for the simplest of them: ultrapowers of $\mathbb{R}$. (key-words: hyperreals, non standard analysis)

-Superreal fields are also proper extensions of $\mathbb{R}$. They are more general than hyperreal fields (and they are extensions of hyperreal fields), and they also contain infinitesimals. They were introduce mainly to enlarge the framework in which several open problems regarding Banach algebras could be anwswered.

-Finaly $No$ is the only ordered Field (up to isomorphism) such that for any subsets $A,B$ of $No$ with $A < B$, there is an element $x$ of $No$ such that $A < x$ and $x < B$. It is a beautiful inductive construction by John Conway (other beautiful constructions exist, see Wikipedia) where there often is a way to define natural extensions of classical operations. It is also universal in the sense that every ordered field and every real closed Field embed in it. There are other Fields satisfying this. $No$ contains in a way "all kinds of infinitesimals"; but this only means that however you embed an ordered field $k$ in it, it will be contained in subfields of $No$ with infinitesimals a lot lower than that of $k$. However, there are proper non isomorphic subFields of $No$ with infinitesimals as low as that of $No$.

The fact that every ordered field embeds, that the ordinal monöid embeds in $No$ combined with the fact that one can define interesting subfields, maps, subrings in $No$ makes it a perhaps promising object of study.

If $k$ is an ordered field, you can always create a proper extension $k(X)$ of $k$ seen as the field of fractions with one indeterminate $X$ where the positive elements are fractions $F(X) = \frac{P(X)}{Q(X)}$ such that $sign(P(X)) = sign(Q(X))$. And $sign(\sum \limits_{i=0}^d x_iX^i) = sign(x_d)$.

If you do that with the field of surreal numbers $No$, you get an extension $No(X)$. $No(X)$ is not real closed so it doesn't embed in $No$, and you can't define a real closure for $No(X)$ but you can still keep going and define $No(X_1,X_2)$, and so on...

$\mathbb{R}$ is useful in mathematics in great part because (modern) mathematics use $\mathbb{R}$ as an elementary component to build more abstract objects. It is also (and this is historicaly important) the good field to model physical space (even though it is likely that an ultrafinitist point of view would be most accurate, $\mathbb{R}$ is at least a good homogenous approximation of space with every existence theorem we need). The are are also arithmetic statements about $\mathbb{Q}$ easier to prove in $\mathbb{R}$ once the general theorems have been established: the density of $\{x^n \ | \ x \in \mathbb{Q}_+\}$ in $\mathbb{Q}_+$ for instance.

I have read there are few theorems proven in IST (an instance of non-standard analysis) that are yet to be proven using regular analysis in ZFC.

I think $No$ is still at an early age of study (it isn't even forty years old), I don't know of any application of this study to other domains, even in field-theory (but I am not omniscient). [edit] $No$ is for now rather studied for itself (just like one might want to study complex numbers for themselves), and for its role as a "monster model" or sometimes just universal model of several well-behaved theories.

  • $\begingroup$ Thank you for your answer that seem to fit many of many questions (+1). I've to re read and better study it. But what about my last question? Such extensions of real numbers have some utility in other branch of math.? I.e. there is some (relevant) result in analysis or algebra or functional theory or other that can be find only using such fields? $\endgroup$ – Emilio Novati Oct 8 '15 at 18:56
  • $\begingroup$ @Emilio Novati: I have edited with some additional info. I hope specialists of the topic can tell you more about applications. $\endgroup$ – nombre Oct 12 '15 at 22:04
  • $\begingroup$ Maybe I should pose this as a separate question, but: the reals are the unique (to iso) linear, dense, unbounded, complete set with a countable dense subset. In what ways are the {sur|super|hyper}reals not counterexamples? By which I mean, for example, are the surreals not complete in some sense? Or do they not qualify solely because they can't fit into a set? $\endgroup$ – Trixie Wolf Sep 11 '17 at 12:43
  • $\begingroup$ Superreal and hyperreal fields do not have a countable dense subset and they are not complete as linear orders, and they might have proper dense field extensions. This is the same for surreals, though they can be deemed "complete" by restricting some conditions to set-sized classes, for instance they are "connected" in the "topology" whose open subclasses are set-sized unions of intervals. $\endgroup$ – nombre Sep 11 '17 at 21:24
  • $\begingroup$ Aside: a nonarchimedean field really is determined by its algebra of infinitesimals, since every element can be written as a ratio of infinitesimals and the equality, arithmetic, and ordering of such ratios can be expressed entirely in terms of the same for the infinitesimals. $\endgroup$ – Hurkyl Dec 23 '17 at 10:01

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