The surreal-numbers tag has no usage guidance.

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### Is $\omega^\frac{1}{\omega} > n \forall n \in \mathbb{N}$?

I was thinking about $log(\omega)$ which appears to be $\{\mathbb{N}|\omega^{\frac{1}{n}}\}_{n\in\mathbb{N}}\stackrel{?}{=}\omega^\frac{1}{\omega}$. Intuitively, there's the idea that, if the highest ...

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106 views

### Extension to real number system [closed]

Suppose you have equation involving a number $s$
$s^2+ 1 = 0$,
to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit.
Now suppose you have equation ...

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**0**answers

273 views

### Is the universality of the surreal number line a weak global choice principle?

I'd like to consider the principle asserting that the surreal
number line is universal for all class linear orders, or in other
words, that every linear order (including proper-class-sized)
linear ...

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votes

**1**answer

117 views

### Ordinals which embed in surreal subfields

If $k$ is an ordered field, the least ordinal $s(k)$ which doesn't embed in $(k,<)$ is regular. This is because every interval of an ordered field embeds in every infinite interval so given a ...

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### Surreal compactness

In a comment here, Joel David Hamkins said:
...I think perhaps every set-sized open cover of a bounded interval in the surreals has a finite subcover, but there are proper class open covers with ...

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**1**answer

310 views

### Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this ...

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502 views

### Nice sign-expansions of special surreal numbers

What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?
I can think of more than one natural way to ...

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237 views

### Nimbers and Surreal Numbers [closed]

I have been researching Combinatorial Game Theory. One common theme is the assignment of values to games in order to classify the game as a win for a specific player. One such way is class of surreal ...

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**1**answer

366 views

### Is it possible to evaluate Connect 4 positions with Combinatorial Game Theory?

The surreal numbers in Combinatorial Game Theory only work for certain classes of games (e.g. they must satisfy normal play convention). This rules out even reasonable games with fairly ...

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**3**answers

245 views

### First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?)

It is known that locally one can ``code'' any set in the von Neumann universe $V$ by a set of ordinals. But can one do this globally? In other words, is there a first-order definable bijection ...

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**1**answer

275 views

### Transcendence degree of the surreals over the subfield generated by the ordinals

Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...

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685 views

### Does this construction yield the surreal numbers?

There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals.
First given such a field one may consider rational functions over that field ...

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**0**answers

378 views

### More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...

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**0**answers

131 views

### An application of surreal numbers towards fast-growing ordinal functors?

The surreal numbers $\mathbb{SN}$ form a class of numbers introduced by J.H. Conway, which behave as an ordered field (even if technically it is not a set). In particular, Conway showed that every ...

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**1**answer

388 views

### Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, ...

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174 views

### Is a certain group, derivable from the surreal numbers, isomorphic to the surreal numbers?

Let's treat $\mathbf{No}$ as a group under addition, and forget its field structure for a little bit.
I will define a "maximally Archimedean subgroup" of $\mathbf{No}$ as a subgroup which is
...

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878 views

### Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig identities) will still ...

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votes

**1**answer

618 views

### Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.
Part 1 is about foundations. Much of the ...

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votes

**1**answer

1k views

### Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...

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**1**answer

870 views

### Integration in the surreal numbers

In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace ...

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**1**answer

437 views

### Definable map from all the ordinals to the surreal numbers with a dense image?

I'm trying to understand analogies and disanalogies between ${\Bbb R}$, the reals numbers, and ${\bf No}$, the surreal numbers.
${\Bbb R}$ admits countable dense sets such as the rationals. This ...

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votes

**3**answers

479 views

### Uniformizing the surcomplex unit circle

Is the multiplicative Group of surcomplex numbers of modulus 1 isomorphic to the additive Group of the surreal numbers modulo the sub-Group of surreal integers? And, do Norman Alling's surreal ...

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**1**answer

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### Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...

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### Constructing the surreal numbers as iterated Hahn series

A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with ...

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**4**answers

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### A question about J.H. Conway's SURREAL NUMBERS

My quesion is: What set theory are the mathematicians who are developing the theory of
these numbers working in-or are they, in fact, working outside any of the standard set
theories?. Each surreal ...

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votes

**3**answers

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### Surreal Numbers and Set Theory

Hello,
I looked through MathOverflow's existing entries but couldn't find a satisfactory answer to the following question:
What is the relationship between No, Conway's class of surreal numbers, and ...

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votes

**3**answers

1k views

### Surreal Numbers as Inductive Type?

Prompted by James Propp's recent question about surreal numbers, I was wondering whether anyone had investigated the idea of describing surreal numbers (as ordered class) in terms of a universal ...

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**5**answers

3k views

### Where do surreal numbers come from and what do they mean?

I know about Conway's original discovery of the surreal numbers by way of games,
as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior
of real-valued functions, leading ...