# Questions tagged [surreal-numbers]

For questions about the surreal numbers, an inductively constructed ordered field that naturally contains all ordinal numbers.

62 questions
11 views

### Conway Notation for Large Countable Ordinals

I have not previously seen anything online that dives deeply into On: In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of ...
67 views

### Prove that a surreal number is born in a finite stage if and only if it is of the form $\frac m{2^n}$.

We define surreal numbers here. My attempt is to first prove this lemma: Lemma 1. Suppose in the $n$th stage, we have already constructed 2 surreals $a<b$, with no other surreals constructed ...
29 views

### $\omega$th iteration of Cayley-Dickson construction

The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) -...
64 views

### Proof that “$\uparrow$ is the unique solution of $tiny(G) = G$”

Tiny & miny games can be defined as: $$tiny(G) = \{0||0|-G\}$$ $$miny(G) = -tiny(G) = \{G|0||0\}$$ From the Wikipedia page for tiny and miny: Similarly curious, mathematician John Horton ...
73 views

### Is multiplication of games that are equivalent to numbers well-defined?

It's well-known that if you take the definition of surreal multiplication and attempts to generalize it to all games, the result is not well-defined, in that it does not respect equivalence of games. ...
92 views

### Improper integrals over the reals and surreal numbers

Is it possible to assign improper integrals over the reals a surreal value in a consistent way? Are there any papers available on this? Note that I am not inquiring about formalizing integration over ...
97 views

### How do surreal numbers relate to real numbers? [closed]

I had the impression that surreal numbers were a subset of reals, being the smallest possible interval away from any other number you could specify. Now, after reading the book, “Surreal Numbers”, it ...
75 views

### Proof that $x+(-x)=0$ for surreal numbers

This is from Conway's on numbers and games: $x+(-x)=0$. We have to show $x+(-x)\geq 0$ and $x+(-x )\leq 0$. If say $(x+(-x))\ngeq 0$, we should have some $(x+(-x))^R\leq 0$, that is $x^R+(-x)\leq 0$ ...
39 views

204 views

360 views

### Are there countably infinte surreal number?

I was thinking about surreal numbers, and about how you can display them like a binary tree (like this) and that since they can be displayed as a binary tree shouldn't there be only countably infinite ...
78 views

### what kind of integral domain do the non-infinite surreals form?

https://en.wikipedia.org/wiki/Integral_domain mentions the following chain of inclusions: Principal Ideal domains $\subset$ Unique Factorization domains $\subset$ GCD domains $\subset$ Integrally ...
169 views

### Algebraic closure vs Real closure

I have proved that the surreal numbers have the properties of a real closed field. Now I should be able to explain what the importance of this real closure is. unfortunately I do not have a background ...
142 views

### Are all nimbers included in the surreals?

I guess the question says it all. The **nimber* (https://en.wikipedia.org/wiki/Nimber) concept, sometimes called "Sprague-Grundy numbers" embodies the "values" of positions in impartial games which ...
97 views

### Completing ordered Fields

How do these two forms of completion behave (in NBG) when fields are authorized to be proper classes? $(i)$: Every ordered field has a real closed, algebraic extension. $(ii$): Every ordered field ...
60 views

### Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $< \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
655 views

### Why is epsilon not a rational number?

I was wondering why epsilon, the smallest positive number, isn't a rational number. I was watching a video a few days ago about surreal numbers, and I've learned that, in the field of surreal numbers, ...
121 views

### Surreal numbers whose final segment is an integer.

For every ordinal $\alpha$, define $a_{\alpha} = \{0\} \ | \ \{a_{\beta} \ | \ \beta < \alpha\}$. In Harry Gonshor's approach of surreals where they are $(+,-)$ sequences of ordinal domain, \$a_{\...
431 views

### Relatively tight upper and lower bounds for surreal numbers

As is well, known, the surreal numbers have gaps. As far as I understand, this means that a set of surreal numbers will not always have a supremum or infimum in the surreal numbers. So I thought ...
474 views

### More than the real numbers: hyperreals, superreals, surreals …?

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 ''...
112 views

### Automorphism group of the class of surreal numbers

Do we know the group (Group) of automorphisms of the ordered Field of surreal numbers? I feel the different ways to see the surreal numbers should provide us with several ways to define interesting ...
181 views

### Decoding the sign expansion of surreal numbers

One way to represent surreal numbers is the sign expansion. Now Wikipedia describes how to compare them, how to convert them to the standard representation of left/right sets, how to negate them, and ...