# Questions tagged [surreal-analysis]

The surreal-analysis tag has no usage guidance.

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### Surreal numbers and the ultrafilter lemma

In this question, I asked how to interpret a historical claim made by Conway regarding the potential, if not ideal, use of the surreal numbers for nonstandard analysis. In the comments and answers it ...

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### Interpreting Conway's remark about using the surreals for non-standard analysis

In Conway's "On Numbers And Games," page 44, he writes:
NON-STANDARD ANALYSIS
We can of course use the Field of all numbers, or rather various small
subfields of it, as a vehicle for the ...

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### Can one represent divergent integrals or germs at infinity with surreal numbers?

I have been disliking the theory of surreal numbers for a while, but let's test it.
So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\...

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### An extension of the mean-value theorem for integrals? [closed]

The mean value theorem for integrals states that if $f$ and $g$ are continuous on $[a, b]$ and $g$ never changes sign on $[a, b]$, then there exists some $c\in [a, b]$ such that
$$\int_{a}^{b} f(x)g(...

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### Surreal numbers and the Axiom of Choice

In ZFC and its conservative extension NBG, it can be shown that every ordered field embeds into the surreal numbers.
How much choice is needed to prove this?
Without choice, what is a simple example ...

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### Algebraic Geometry Over the Surreal and Surrcomplex Numbers

I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?

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### Genetic construction of roots of surreal polynomials

In On Numbers And Games, Conway uses the term "genetic" for definitions of operations on surreal numbers that are inductive in terms of their options. His definitions of addition and multiplication ...

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### Can a game be an option of itself?

My question is, can a game contain itself as an option? and can it be a surreal number? For example $A=\{A|\}$ or $B=\{C|B\}$ where $C$ is a surreal number.
from the point of view of games, it is ...

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### The surreal version of $e$

For a sequence $(x_{\alpha})$ of surreal numbers indexed by the set of all ordinal numbers, we say that $\lim x_{\alpha}=l$ ($l$ is a surreal number) if for each surreal $\epsilon>0$, there exists ...

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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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### 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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### Theory of C* algebras over other fields than the complex numbers

How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...

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### 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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### 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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### Surreal numbers vs. non-standard analysis

What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in ...

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### Characterizing the surcomplex numbers

Conway showed that the Field of surreal numbers ("${\bf No}$")
is the maximal totally ordered Field.
Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is
the universally ...