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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 ...
Mike Earnest's user avatar
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24 votes
2 answers
8k views

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 ...
Nathan Reed's user avatar
21 votes
4 answers
2k views

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 ...
celtschk's user avatar
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15 votes
1 answer
1k views

Surreal numbers without the axiom of infinity

Let $ZF^\times$ denote the set of axioms of Zermelo-Fraenkel set theory without the axiom of infinity. The set $V_\omega$ of all hereditarily finite sets is a model of $ZF^\times$, and $Ord^{V_\omega}=...
Asaf Karagila's user avatar
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15 votes
3 answers
336 views

Modeling numbers with vectors of vectors?

I stumbled upon the strange representation of integers where $$8=\langle\langle0,\langle0^{\infty}\rangle,0^{\infty}\rangle,0^{\infty}\rangle$$ I'll try explain the representation in a natural way. ...
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14 votes
2 answers
2k views

Surreal and ordinal numbers

Is there a surjective map between the (class of) ordinal numbers On and the set No (Conway's surreal numbers) and is it constructable, In Conway's system we have for example: $\omega_0 = < 0,1,2,3,...
user avatar
14 votes
1 answer
673 views

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 ...
E8xE8's user avatar
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13 votes
1 answer
2k views

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)...
pregunton's user avatar
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12 votes
2 answers
2k views

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 ...
SK19's user avatar
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12 votes
1 answer
203 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. ...
Harry Altman's user avatar
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11 votes
5 answers
4k views

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 ...
Presh's user avatar
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11 votes
1 answer
1k 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 ''...
Emilio Novati's user avatar
9 votes
3 answers
1k views

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 ...
RothX's user avatar
  • 1,681
9 votes
3 answers
923 views

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, ...
kuzzooroo's user avatar
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9 votes
1 answer
582 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 ...
Mark Fischler's user avatar
9 votes
2 answers
2k views

Proof of Conway's "Simplicity Rule" for Surreal Numbers

A "number" in the sense of Combinatorial Game Theory is a game $G = \{ a,b,c,\dots | \; d,e,f,\dots \}$ such that $a,b,c < d,e,f$. Then our game is between the left and right options: $$ a,b,c &...
cactus314's user avatar
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8 votes
2 answers
486 views

Surreal arithmetic with $\frac{1}{2}\omega$

In the final two chapters of Knuth's Surreal Numbers, both the world of multiplication and infinite/infinitesimal numbers are introduced. The basic ideas of both of these make sense to me, but I'm ...
Pat Muchmore's user avatar
8 votes
2 answers
391 views

Does every total order embed into the surreal numbers?

The surreal numbers are the largest ordered field, and have the unique property that every ordered field is isomorphic to a subfield of the surreal numbers. Do they also have the property that every ...
Mike Battaglia's user avatar
7 votes
2 answers
1k views

Integral of a function defined in the set of Surreal Numbers

Given ${\{C}\}\ $ the set of all the $Surreal\ numbers$, is it possible to define the integral: $$\int_a^b{dxf(x)}$$where $$a\in{\{C}\},b\in{\{C}\},x\in{\{C}\}$$ Thanks
Riccardo.Alestra's user avatar
7 votes
1 answer
824 views

A question about something in Conway's "On Numbers and Games"

In the book mentioned in the title, which deals with (among other things), Conway's "surreal numbers", there is a small section (pp. 37-38) where the "gaps" in the surreal number line are discussed. ...
The_Sympathizer's user avatar
7 votes
1 answer
164 views

How much arithmetic can we find definably in the surreals?

Playing fast and loose with size issues for simplicity, let $\mathfrak{S}$ be the structure of the surreal numbers equipped with addition, multiplication, and the simplicity order. I'm curious how ...
Noah Schweber's user avatar
7 votes
3 answers
1k 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, ...
Asix's user avatar
  • 555
7 votes
1 answer
226 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 ...
nombre's user avatar
  • 5,117
7 votes
0 answers
88 views

Transfinitely iterating the Puiseux, Levi-Civita, or Hahn series constructions

There are many ways to take some real-closed field and generate a proper extension of it with elements that are infinite and infinitesimal relative to the original field. One well-known example is ...
Mike Battaglia's user avatar
7 votes
0 answers
373 views

is there a p-adic analogue to the surreal numbers?

Because the p-adics are analogous to the reals, I was wondering if there was a structure that was analogous to the surreals in the same way, i.e this structure would have similar quirks/properties to ...
Mettek's user avatar
  • 437
6 votes
2 answers
217 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 ...
nombre's user avatar
  • 5,117
6 votes
1 answer
1k views

Curiosity with surreal numbers

I'm a high school student who is interested in surreal numbers and studying it by myself. Suddenly a weird idea popped in my mind. For example, can we define an event $A$ that has $P(A)=\frac1{\omega}$...
user avatar
6 votes
1 answer
177 views

Does surreal numbers with bounded birthdays form a field?

For a surreal number $X = \{X_L|X_R\}$, its birthday is defined recursively to be the smallest ordinal bigger than the birthdays of all $x \in X_L \cup X_R$. If $X$ can be written as $\{X_L|X_R\}$ in ...
Jacob FG's user avatar
  • 746
6 votes
3 answers
182 views

Is the class of ordinal numbers bounded by a fixed ordinal number really a set when defined through surreal numbers?

At his book "On Numbers and Games", Conway defines ordinal numbers as games which doesn't have right options and whose left options contain only ordinal numbers. Then, fixed an ordinal ...
Anderson Brasil's user avatar
6 votes
2 answers
164 views

What is the “maximal hyperreal field”?

In many SE posts and the Wikipedia article on the surreal numbers I’ve seen references to a “maximal” hyperreal field that’s isomorphic to the surreals. If they’re isomorphic, then why is it that ...
Lave Cave's user avatar
  • 1,159
6 votes
0 answers
177 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 ...
kapaw's user avatar
  • 456
5 votes
2 answers
848 views

Notation for surreal numbers

On the sound of sounding ridiculous, but in the line of "There are no stupid quetsions": Is there a way to express $\omega_1$ (and in general $\omega_k$ with $k >= 1$ as a Conway game (that is $<...
Willem Noorduin's user avatar
5 votes
2 answers
215 views

How does the empty set work in arithmetic of surreal numbers?

I'm working my way through Surreal Numbers by Knuth, and am finding myself a little hung up on the explanation of how addition works. The rule for addition is given as: $$ x + y = ((X_L+y)\cup(Y_L+x),...
Pat Muchmore's user avatar
5 votes
1 answer
117 views

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 ...
Jonathan Lenchner's user avatar
5 votes
1 answer
313 views

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 ...
Lave Cave's user avatar
  • 1,159
5 votes
2 answers
266 views

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 ...
Thomas Benjamin's user avatar
5 votes
2 answers
302 views

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,...
JtSpKg's user avatar
  • 415
5 votes
1 answer
484 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 ...
celtschk's user avatar
  • 43.5k
5 votes
2 answers
568 views

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}, ...
alan2here's user avatar
  • 1,017
5 votes
1 answer
55 views

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 ...
M masa's user avatar
  • 161
5 votes
4 answers
820 views

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 (...
B. Elliott's user avatar
4 votes
2 answers
489 views

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, ...
Johnathan Green's user avatar
4 votes
2 answers
337 views

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 ...
user784623's user avatar
4 votes
1 answer
216 views

Why is $\uparrow+\ast=\{\ast,0|0\}$?

On p. 189 of Conway's ONAG, he describes a position in Hackenbush Hotchpotch as $\{\ast,0\:|\:0\}$ and then says that $\{\ast,0\:|\:0\}=\:\uparrow+\:\ast$. (where $\ast\equiv\{0\:|\:0\}$ and $\uparrow\...
Pat Muchmore's user avatar
4 votes
1 answer
444 views

Can a set of surreal numbers be defined with arbitrary cardinality?

It is my understanding that the surreal numbers form a class rather than a set, because their collection is larger than any set. Thus it would seem to follow that for any cardinality, such as $\...
user avatar
4 votes
1 answer
486 views

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 ...
M. Winter's user avatar
  • 30.1k
4 votes
1 answer
111 views

Surreal numbers ordering and false inequality $1\le0$

Given numeric forms $x =\{ X_L \,|\, X_R\}$ and $y =\{ Y_L \,|\, Y_R\}$ of two surreal numbers we say that $x \le y$ if and only if There is no $x_L \in X_L$ such that $y \le x_L$ and There is no $...
Zeekless's user avatar
  • 1,489
4 votes
1 answer
172 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 ...
Trebor's user avatar
  • 4,776
4 votes
1 answer
319 views

Attempt to define limit of a sequence of surreal numbers

For sake of well-definedness, here we consider only ordinals less than the first uncountable ordinal, $\Omega$. Just like $\infty$ in the notation $\lim_{n→\infty}$ is essentially $\omega$, $\Omega$ ...
Dannyu NDos's user avatar
  • 2,059
4 votes
1 answer
539 views

Completion of surreal numbers

Surreal number field $\mathbf{No}$ is not complete, there are "gaps". Does there exists a completion of it? I know this question depends on axioms of set theory and more, feel free to ...
mz71's user avatar
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