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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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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 ...
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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 ...
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$\omega$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) -...
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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 ...
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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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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 ...
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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 ...
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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 $x^R+(-x)\leq 0$ ...
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Descending Game Condition and Transfinite induction

I am having a bit of trouble understanding the proof of Conway induction. Definition: Descending Game condition: There does not exist an infinite sequence of games $G^i=(L^i,R^i)$ with $G^{i+1}\...
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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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Do surreals prove reals are countable?

If the surreal number $\epsilon = 1/\omega$ is the lower bound of the difference between any two real numbers (since it is smaller than any real number), and there are a countable number of these ...
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Are aleph numbers contained in the set of surreal numbers or hyperreal numbers? [closed]

Do surreal or hyperreal number sets contain any member that's an aleph number?
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What is the CGT value of Misère Hex?

Misère Hex is like Hex, except that the winner and loser are swapped. Instead of trying to make a connection, you are trying to not make a connection. One player will inevitably lose, since when the ...
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Roots of polynomials of $R[x]$ in the surreals

I have this question and I don't know where to even begin. The question is: Let $S$ denote the surreals. Prove or disprove: no polynomial in $R[x]$ has a root in $S \setminus \mathbb{R}$. Help!
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examples of sign expansion from normal forms of surreal numbers

I am reading Conway's "On Numbers and Games" and in chapter 3, I'm a little confused. In the part where normal forms and sign expansions are discussed together, the direction is to use relevant sign ...
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Any references on supercomplex/surcomplex numbers?

I know many text in surreal numbers have went over them but I can't seem to find anything that goes into great detail beyond a small section on them. For instance, in Foundation of Analysis over ...
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How are the cardinalities of infinite sets described in systems where the continuum hypothesis doesn't hold?

As I understand it: In ZFC the continuum hypothesis can neither be proven true nor false, or in other words, a new axiom could be added to ZFC that says the continum hypotheis is true or that it is ...
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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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Must sign expansion definition of surreal addition use induction on the natural sum?

In Gonshor surreals and their arithmatic are defined using sign expansions. Addition of surreals $a,b$ is defined inductively by $a+b = (a_L+b,a+b_L)\mid (a_R+b,a+b_R) $ Where the induction is on ...
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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\...
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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 $\...
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Surcomplex Transfinite Maths

So just curious about the surcomplex numbers... Do they have transfinites like the surreal numbers? For example: ω*i = ? Would that even make sense? Thanks!
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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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Is reverse-order-omega the same as negative omega?

In a previous question, I asked whether or not there was such as thing as negative transfinite numbers, such as negative omega, -ω. I received an answer that suggested ω* was the solution I had been ...
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What is new representatives for well-known values in surreal numbers?! [closed]

I know how to obtain surreal numbers in n-day and I know about <= in surreal numbers axiom 2. We will also discover a lot of new representatives for well-known values. For example : $\{-1|1\}=0$ or ...
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List of equivalence surreal numbers to 4 day?

I can obtain surreal numbers in n-day then I don't want only list of surreal numbers example in 2-day: -2<-1<-1/2<0<1/2<1<2 but I want list of equivalence surreal numbers exemplar 2-...
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Why is not some numbers in nth days?

In surreal numbers : In second day we have 2 and 1/2 and ... but why in third day we don't have {1/2|2} =5/4 ?
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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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What is the difference between the hyperreals and $R^3$?

I just found out about the hyperreals. Here is a visualization of the hyperreals: The problem I have with this image, is that it conveys a very clear intuition, but often times, those intuitions are ...
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Surreal numbers - does the triangle number formula work?

In surreal numbers, is there a solution to: 1+2+3+...+ω? In particular, is the solution ω(ω+1)/2? Intuitively, it seems like it should still work because I can imagine putting these two triangles ...
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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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Integrating a surreal-valued function over a probability space

Is there some agreed definition of the integral of a surreal-valued function over a probability space?
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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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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 ...
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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),...
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Universal linear orders without $\omega_1$ or $\omega_1^*$ subsets.

Let's say that a linear order is $\omega_1$-short if it has no uncountable well-ordered or reverse well-ordered subset. For instance, linear orders satisfying Suslin's condition in his hypothesis are $...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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, ...
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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_{\...
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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 ...
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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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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 ...
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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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Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
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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}, ...