# Questions tagged [surreal-numbers]

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

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### How does Conway's proposed compromise for constructing the real numbers actually work?

My question is about understanding a remark John Conway made in On Numbers and Games (ONAG), where he proposes a method for constructing the real numbers from the rationals. I will have to assume ...
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### What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain: real numbers a hierarchy of infinitesimal ...
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### Are surreal numbers actually well-defined in ZFC?

Thinking about surreal numbers, I've now got doubts that they are actually well-defined in ZFC. Here's my reasoning: The first thing to notice is that the surreal numbers (assuming they are well ...
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### Non-standard measure

Just a bit of a strange question. Modern formulations of probability theory rest upon measure theory. This poses an issue for non-measurable sets. Typically, one simply excludes these sets from the ...
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### Surcomplex numbers and the largest algebraically closed field

It's well known that the surreal numbers $\mathbf{No}$ are the largest ordered "field" (more accurately, they form a proper class with field structure, which is sometimes called a Field with capital F)...
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### Why does the inverse of surreal numbers exist?

Problem I'm working with the book "On numbers and games" from John Conway, first edition from 1976. On page 20 he writes Summary. Numbers form a totally ordered Ring. Note that in view of Theorem ...
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### 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. ...
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### In the surreal numbers, is it fair to say $0.9$ repeating is not equal to $1$?

I find the surreal numbers very interesting. I have tried my best to work through John Conway's On Numbers and Games and teach myself from some excellent online resources. I have prepared a short ...
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### 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 ''...
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### Examples of Surreal Numbers that are only Surreal Numbers?

I was just reading through the construction of the surreal numbers on wikipedia, and I read through some of the examples. I noticed that all of the examples were how certain types of already existing ...
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### Why are the surreals considered "recreational" mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
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### 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 ...
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### Question about the $* = \{0\mid 0\}$ Game in "On Numbers and Games"

Why is the game $* = \{0\mid 0\}$, where $0=\{\;\mid\;\}$ is the empty game, mentioned on p.72 of Conway's "On Numbers and Games", a win for the 1st player? It seems to me that if the 1st ...
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### Can the surreal numbers be completed to form an ordered field?

The surreal number line isn’t Cauchy complete as it’s filled with “gaps”. When constructing the real numbers from the rationals, one could take the ring of all Cauchy sequences and take the quotient ...
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### In the field $No$ of surreal numbers, does $\underbrace {\frac 1 \omega + \frac 1 \omega + ...}_{\omega\text{ times}}= 1?$

The question is in the title. It is known that $\omega$ $\cdot$ $\frac 1 {\omega}$ = 1, but can the expression on the left-hand side be replaced by the infinite sum in the title? If so then by the ...
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### Surreal numbers as generalized Dedekind cuts

From the four postulates of the Dedekind cuts, namely (for (a,b) denoted as the cut, a,b being subsets of the rationals): Every rational number lies in exactly one of the sets a,b, a,b are not empty,...
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### 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 ...
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### Pseudo-Surreal numbers are analogous to?

I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|} I see that pseudo-surreal numbers seem to have an interesting branch of game theory. Still having a form of {x|y}, ...
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### Hyper-extensions of Hom space

We fix an ultrafilter $\mathcal{F}$ of $\mathbb{N}$ which contains the cofinite filter. Let $A,B$ be sets and ${}^{*}A,{}^{*}B$ their hyper-extensions. Then is $${\rm Hom}({}^{*}A,{}^{*}B)$$ equal ...
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### Algorithm for Converting Rational Into Surreal Number

I'm writing a library in Haskell that represents the class of surreal numbers, which like many things can be read about here. I've run into a problem in converting between other classes of numbers (...
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### Could a coordinate space be defined over the surreal numbers?

Self-explanatory. I am aware of coordinate spaces being defined over the field of real and complex numbers, but I've wondered whether you could do something similar with the field of surreals: As in, ...
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### Is $\infty=\sqrt[\Omega]{\omega}$ actually a thing?

In Infinity and the Mind the author claims: Conway derives the weird equation, $\infty=\sqrt[\Omega]{\omega}$, which almost magically ties together potential infinity $\infty$, the simplest actual ...
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### Are the hyperreals emerging at some stage of the surreal construction?

For me What's the difference between hyperreal and surreal numbers? has a not very satifying answer. I always pictured the hyperreals as some subfield of the surreals naturally emerging when you ...
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